{"componentChunkName":"component---src-templates-post-tsx","path":"/machine_learning_math/vectorization/","result":{"data":{"logo":{"childImageSharp":{"gatsbyImageData":{"layout":"fixed","backgroundColor":"#f8f8f8","images":{"fallback":{"src":"/static/e43c6530b7b2b056de6bfee951c9e393/b181d/compasslogo.png","srcSet":"/static/e43c6530b7b2b056de6bfee951c9e393/b181d/compasslogo.png 1920w","sizes":"1920px"},"sources":[{"srcSet":"/static/e43c6530b7b2b056de6bfee951c9e393/e02c9/compasslogo.webp 1920w","type":"image/webp","sizes":"1920px"}]},"width":1920,"height":472}}},"mdx":{"body":"const _excluded=[\"components\"];function _extends(){_extends=Object.assign||function(target){for(var i=1;i<arguments.length;i++){var source=arguments[i];for(var key in source){if(Object.prototype.hasOwnProperty.call(source,key)){target[key]=source[key];}}}return target;};return _extends.apply(this,arguments);}function _objectWithoutProperties(source,excluded){if(source==null)return{};var target=_objectWithoutPropertiesLoose(source,excluded);var key,i;if(Object.getOwnPropertySymbols){var sourceSymbolKeys=Object.getOwnPropertySymbols(source);for(i=0;i<sourceSymbolKeys.length;i++){key=sourceSymbolKeys[i];if(excluded.indexOf(key)>=0)continue;if(!Object.prototype.propertyIsEnumerable.call(source,key))continue;target[key]=source[key];}}return target;}function _objectWithoutPropertiesLoose(source,excluded){if(source==null)return{};var target={};var sourceKeys=Object.keys(source);var key,i;for(i=0;i<sourceKeys.length;i++){key=sourceKeys[i];if(excluded.indexOf(key)>=0)continue;target[key]=source[key];}return target;}/* @jsxRuntime classic */ /* @jsx mdx */const _frontmatter={\"layout\":\"post\",\"title\":\"Vectorization Explained, Step by Step\",\"author\":[\"Lari Giba\"],\"tags\":[\"Machine Learning Math\"],\"image\":\"img/Vectorization/Vectorization_Thumbnail.png\",\"date\":\"2021-03-06T18:32:05.647Z\",\"dateModified\":\"2021-03-06T18:32:05.647Z\",\"changelog\":[[\"2021-12-28T13:18:41.297Z\",\"Fixed some issues in the math examples. Thanks to stumpeschach for pointing them out!\"]],\"draft\":false,\"imgCaption\":\"Background image by Joel Filipe\",\"imgSource\":\"https://unsplash.com/photos/ZMRMFULofus\",\"timeToRead\":7,\"uniquePostId\":4,\"excerpt\":\"Vectorization is one of the most useful techniques to make your machine learning code more efficient. In this post, you will learn everything you need to know to start using vectorization efficiently in your machine learning projects.\",\"dependencies\":[],\"proficiencyLevel\":\"Beginner\"};const makeShortcode=name=>function MDXDefaultShortcode(props){console.warn(\"Component \"+name+\" was not imported, exported, or provided by MDXProvider as global scope\");return mdx(\"div\",props);};const ToolTip=makeShortcode(\"ToolTip\");const ConditionalLazyPlot=makeShortcode(\"ConditionalLazyPlot\");const layoutProps={_frontmatter};const MDXLayout=\"wrapper\";return function MDXContent(_ref){let{components}=_ref,props=_objectWithoutProperties(_ref,_excluded);return mdx(MDXLayout,_extends({},layoutProps,props,{components:components,mdxType:\"MDXLayout\"}),mdx(\"h2\",{\"id\":\"outline\"},`Outline`),mdx(\"p\",null,`In this post, we will go over what vectorization really is, what the benefits of vectorizing your\ncode are, and how you can do so. You do not need to have any specific knowledge prior to reading this post,\nbut it will be very helpful if you are familiar with linear regression and the\nmean squared error since we will use those as an example. If you are not, I recommend\nyou read the post `,mdx(\"a\",{parentName:\"p\",\"href\":\"/machine_learning_models/linear_regression\"},`Linear Regression Explained, Step by Step`),`,\nwhere you will learn everything you need to know about those two. This post and the linear regression post,\nin particular, go hand in hand, as the examples in this post are taken directly\nfrom linear regression itself. We will also use the same dataset and terminology that was used in the post about\nlinear regression, so even if you are familiar with linear regression, you might still find it helpful\nto just quickly skim through the linked post.`),mdx(\"h2\",{\"id\":\"the-main-idea\"},`The main idea`),mdx(\"p\",null,`If you’ve implemented a few machine learning algorithms on your own or if you perform many calculations\nin your programs, you know how long those calculations can sometimes take.\nLet’s say we have a dataset containing 100.000 data points and we want to perform some sort of operation\non every single one of those points. If we use a regular loop for this, things will get\nvery inefficient very quickly. If we can instead `,mdx(\"em\",{parentName:\"p\"},`vectorize`),` our operation and just perform\none (or a few) large vector or matrix operation using something like `,mdx(\"em\",{parentName:\"p\"},`NumPy`),`,\nour code will run `,mdx(\"em\",{parentName:\"p\"},`a lot`),` faster`,mdx(ToolTip,{trigger:\"*\",mdxType:\"ToolTip\"},mdx(\"a\",{target:\"_blank\",rel:\"noopener noreferrer nofollow\",href:\"https://towardsdatascience.com/how-fast-numpy-really-is-e9111df44347#:~:text=As%20the%20array%20size%20increase,also%20frees%20the%20memory%20faster.\",rel:\"noopener noreferrer nofollow\"},`Here`),` is an interesting article comparing the two. Take a special look at the “Dot product”-section.`),`.\nOk, so now we know why vectorizing our operations might be a good idea. But how can we do so?\nLet’s look at two examples.`),mdx(\"h2\",{\"id\":\"vectorization\"},`Vectorization`),mdx(\"h3\",{\"id\":\"example-1-linear-function\"},`Example 1: Linear Function`),mdx(\"p\",null,`Let’s say we want to use linear regression to predict the price of a house based on the number\nof bedrooms it has. We could create a linear function like this one:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nf(x) = mx+b\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.66666em\",\"verticalAlign\":\"-0.08333em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`)))))),mdx(\"p\",null,`where x is the number of bedrooms in a house. Ok.\nNow let’s say we want to predict the price of a house based on the number of bedrooms,\nthe year it was built in and the average house price in the city the house stands in.\nThen our function\nwould look more like this:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`2`)),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`3`)),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`m`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`2`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`m`),mdx(\"mn\",{parentName:\"msub\"},`2`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mn\",{parentName:\"msub\"},`3`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`m`),mdx(\"mn\",{parentName:\"msub\"},`3`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`f(x_1,x_2,x_3) = x_1 \\\\cdot m_1 + x_2 \\\\cdot m_2 + x_3 \\\\cdot m_3 + b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`3`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59445em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.73333em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59445em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.73333em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59445em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`3`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.73333em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`3`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`)))))),mdx(\"p\",null,`Things got less pretty rather quickly..\nSo to make our lives easier we will vectorize our initial equation!\nThere are a couple of steps we need to take in order to vectorize our equation.\nFirst, we rename our `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`m`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`m`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`))))),` to `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\theta_1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\theta_0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`. So instead of writing`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nf(x) = mx+b\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.66666em\",\"verticalAlign\":\"-0.08333em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`)))))),mdx(\"p\",null,`we now write`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`f(x) = \\\\theta_1x + \\\\theta_0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"p\",null,`This will all make sense in a moment, just bear with me for now.\nThen we change up the notation of our\nfunction a little bit. We now write:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nf(x) = \\\\boldsymbol{\\\\theta} \\\\cdot \\\\mathbf{x}_b\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"p\",null,`In this step, we rewrote our function to now use vectors instead of scalars. In other words,\ninstead of multiplying a bunch of individual numbers together, we now only need to multiply two vectors.\nWe define `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\boldsymbol{\\\\theta}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` like this:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`)),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mi\",{parentName:\"mstyle\"},`x`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\n\\\\boldsymbol{\\\\theta} = \\\\left[\\\\begin{array}{c}\n\n    \\\\theta_0 \\\\\\\\\n    \\\\theta_1 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n,\n\n\\\\textbf{x}_b = \\\\left[\\\\begin{array}{c}\n\n    1 \\\\\\\\\n    x \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.40003em\",\"verticalAlign\":\"-0.95003em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`[`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`]`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.40003em\",\"verticalAlign\":\"-0.95003em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`[`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`]`)))))))),mdx(\"p\",null,`where `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`x`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`x`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`))))),` is our number of bedrooms for a given house.\nLet’s first check if this is equal to our initial definition.`),mdx(\"p\",null,`If we calculate the dot product of our two vectors `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\boldsymbol{\\\\theta}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`,\nwe get:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`)),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`1`),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mi\",{parentName:\"mrow\"},`x`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\n\\\\boldsymbol{\\\\theta} \\\\cdot \\\\textbf{x}_b = \\\\theta_0 \\\\cdot 1 + \\\\theta_1 \\\\cdot x = \\\\theta_0 + \\\\theta_1 \\\\cdot x\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.72777em\",\"verticalAlign\":\"-0.08333em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.43056em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`)))))),mdx(\"p\",null,`As we can see, our function remains unchanged!`),mdx(\"p\",null,`But this might still look a bit confusing. Let’s go through the changes step by step.`),mdx(\"p\",null,`We rewrote our `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\theta_0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\theta_1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` into the vector `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"bold-italic\"},`θ`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\boldsymbol{\\\\theta}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\"}},`θ`))))))),`. So far so good.\nNext up we rewrite our input x into the vector `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`. Ok, but what is this 1 doing in our `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`?\nIsn’t our x supposed to be the number of our bedrooms?\nSince we always want to add our bias term `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`))))),` (now `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\theta_0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.84444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`) without multiplying it\nwith anything (remember: our function looks like this: `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`mx+b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.66666em\",\"verticalAlign\":\"-0.08333em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`))))),`), we need to account for that.\nBy adding a 1 to our x-vector, we make sure that our bias-term is then multiplied by this 1,\nand therefor always added as is inside of our equation.\nThis is also the reason why, in many books and tutorials, you will see people\nadding a 1 to their parameter vector. This is to account for the bias-term.\nNow you could write out the bias-term separately and just not add the 1 to your x-vector,\nbut this way, our equation looks a bit neater on the surface. Both notations are used in practice,\nso now you know both sides of the coin! We also add this little `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mrow\",{parentName:\"msub\"}),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.486108em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` to our `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},`x`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.44444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)))))),` to signal\nthat we accounted for the bias-term already and to also differentiate this from any\ninput we might also call “x”. So if you see an `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`, remember that it means “x, but\nwith a 1 added to account for the bias term”.`),mdx(\"h3\",{\"id\":\"example-2-mse\"},`Example 2: MSE`),mdx(\"p\",null,`So a non-vectorized definition of the MSE might look like this:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`M`),mdx(\"mi\",{parentName:\"mrow\"},`S`),mdx(\"mi\",{parentName:\"mrow\"},`E`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"mi\",{parentName:\"mrow\"},`b`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},`1`),mdx(\"mn\",{parentName:\"mfrac\"},`7`)),mdx(\"munderover\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"munderover\"},`∑`),mdx(\"mrow\",{parentName:\"munderover\"},mdx(\"mi\",{parentName:\"mrow\"},`i`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mn\",{parentName:\"mrow\"},`1`)),mdx(\"mn\",{parentName:\"munderover\"},`7`)),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mi\",{parentName:\"msub\"},`i`)),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`i`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\"},`b`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},`)`),mdx(\"mn\",{parentName:\"msup\"},`2`)))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nMSE(m,b) = \\\\frac{1}{7}\\\\sum_{i=1}^7{(y_i-(mx_i+b))^2}\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10903em\"}},`M`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`S`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`E`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"3.0787820000000004em\",\"verticalAlign\":\"-1.277669em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen nulldelimiter\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"mfrac\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.314em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.23em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}})),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.677em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose nulldelimiter\"})),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mop op-limits\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.8011130000000004em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.872331em\",\"marginLeft\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel mtight\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.050005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mop op-symbol large-op\"},`∑`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.3000050000000005em\",\"marginLeft\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`7`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.277669em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8641079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))))))))))))),mdx(\"p\",null,`There is nothing wrong with that definition, but if we wanted to translate our definition\ninto code, what we would probably do is create a loop that sums up each of the individual residuals.\nThe code could look similar to this:`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`def mse_non_vectorized(data,function):\n    sosr = 0 # sum of squared residuals\n    for (x,y) in data:\n        residual = y - function(x)\n        sosr += residual**2\n    mse = sosr / len(data)\n    return mse\n`)),mdx(\"p\",null,`Now, for our small dataset of 7 points that is no big deal, but imagine we had 100.000 data points\n(which is not uncommon today). Every time we wanted to compute our MSE, we would have to perform\n100.000 operations through our loop. We can do better than this!\nSo let’s vectorize our MSE!`),mdx(\"p\",null,`For this, let’s use an exemplary dataset with two features: the number of bedrooms in a house\nand the price for that house. The dataset looks like this:`),mdx(ConditionalLazyPlot,_extends({},markerData,{mdxType:\"ConditionalLazyPlot\"}),mdx(\"p\",null,mdx(\"span\",{parentName:\"p\",\"className\":\"gatsby-resp-image-wrapper\",\"style\":{\"position\":\"relative\",\"display\":\"block\",\"marginLeft\":\"auto\",\"marginRight\":\"auto\",\"maxWidth\":\"599px\"}},`\n      `,mdx(\"span\",{parentName:\"span\",\"className\":\"gatsby-resp-image-background-image\",\"style\":{\"paddingBottom\":\"67.19999999999999%\",\"position\":\"relative\",\"bottom\":\"0\",\"left\":\"0\",\"backgroundImage\":\"url('data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAANCAYAAACpUE5eAAAACXBIWXMAABYlAAAWJQFJUiTwAAABdklEQVQ4y5VT2W6DMBAkNAcEzB1ic4W7NJWa/v/PTbXGUIJQj4eR1/Z6mN0dtL2bQmdCgvEGQTYgyAf42Ssc3uLFSUA5tG5hvBMSFGt0aEYljLCYiXcEe4wp6SfQ+1NY4hTeZKztbI4gf4MrRjWHXwjW0BmHLQawZIBuc2j2tYafDbAu1VzC36FKZkJCluwlPaLbHeeo/Bfh1LN1+ZqbdBDtAw5voK8ID172nUwq1D2dm3Gz+SGNJknTpaEsFZIC6s9EZvFeNp8GdfQL+OWHmvRaoeiQtA85lGmqRGBearD0Ls8o0RY9jEgRBgWC6rGtkCxDgzHDAntVIhGSDSzezYRGVOHo55u+fFJItvHSDp5o4Zef8pFMoGQmxj46Cdz8XaqmeNnbtc206U8hC1hxhYM3XhhBIfeT2Y0gx9HPZEyELK5xVLnnSwkzUsZeW2UuRU3TjusR10b+irI9UfmU+1TyFCzXZUy2IpATvKSDl/ZS+Tp32n8BxlI3zVlwiPsAAAAASUVORK5CYII=')\",\"backgroundSize\":\"cover\",\"display\":\"block\"}}),`\n  `,mdx(\"picture\",{parentName:\"span\"},`\n          `,mdx(\"source\",{parentName:\"picture\",\"srcSet\":[\"/static/008c89723e2962473c6e6c4bc8c82093/601b1/data_plot.webp 500w\",\"/static/008c89723e2962473c6e6c4bc8c82093/b44a7/data_plot.webp 599w\"],\"sizes\":\"(max-width: 599px) 100vw, 599px\",\"type\":\"image/webp\"}),`\n          `,mdx(\"source\",{parentName:\"picture\",\"srcSet\":[\"/static/008c89723e2962473c6e6c4bc8c82093/0eb09/data_plot.png 500w\",\"/static/008c89723e2962473c6e6c4bc8c82093/1906d/data_plot.png 599w\"],\"sizes\":\"(max-width: 599px) 100vw, 599px\",\"type\":\"image/png\"}),`\n          `,mdx(\"img\",{parentName:\"picture\",\"className\":\"gatsby-resp-image-image\",\"src\":\"/static/008c89723e2962473c6e6c4bc8c82093/1906d/data_plot.png\",\"alt\":\"Exemplary Dataset\",\"title\":\"Exemplary Dataset\",\"loading\":\"lazy\",\"decoding\":\"async\",\"style\":{\"width\":\"100%\",\"height\":\"100%\",\"margin\":\"0\",\"verticalAlign\":\"middle\",\"position\":\"absolute\",\"top\":\"0\",\"left\":\"0\"}}),`\n        `),`\n    `))),mdx(\"p\",null,`In Python, we can implement our dataset as a simple list:`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`data = [(1, 80), (2, 170), (3, 100), (3, 220), (4, 200), (4, 270), (5, 500)]\n`)),mdx(\"p\",null,`Let’s also create an exemplary linear function so we can calculate our MSE:`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`def f(x):\n    return 60*x #number of bedrooms * 60k$ = price of the house\n`)),mdx(\"p\",null,`If we use our non-vectorized implementation of the MSE, we get:`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`>>>mse_non_vectorized(data,f)\n53400\n`)),mdx(\"p\",null,`Ok, now let’s vectorize it! First, let’s make some small changes.\nLet’s reshape our individual data values (1,80), (2,270) and so on into two vectors `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},`x`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.44444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)))))),` and `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},`y`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{y}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.63888em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`y`)))))),`.\nThis will be useful later on when we redefine our MSE.\nThose vectors look like this:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mtext\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},`x`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`2`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`3`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`3`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`4`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`4`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`5`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"mtext\",{parentName:\"mrow\",\"mathvariant\":\"bold\"},`y`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`80`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`170`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`100`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`220`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`200`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`270`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`500`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x} =\n\\\\left[\\\\begin{array}{c}\n\n    1 \\\\\\\\\n    2 \\\\\\\\\n    3 \\\\\\\\\n    3 \\\\\\\\\n    4 \\\\\\\\\n    4 \\\\\\\\\n    5 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n,\n\n\\\\textbf{y} =\n\\\\left[\\\\begin{array}{c}\n\n    80 \\\\\\\\\n    170 \\\\\\\\\n    100 \\\\\\\\\n    220 \\\\\\\\\n    200 \\\\\\\\\n    270 \\\\\\\\\n    500 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.44444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`y`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`8`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))))))))),mdx(\"p\",null,`With this, we can now redefine our MSE like this:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`M`),mdx(\"mi\",{parentName:\"mrow\"},`S`),mdx(\"mi\",{parentName:\"mrow\"},`E`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"mi\",{parentName:\"mrow\"},`b`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},`1`),mdx(\"mn\",{parentName:\"mfrac\"},`7`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"munderover\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"munderover\"},`∑`),mdx(\"mrow\",{parentName:\"munderover\"},mdx(\"mi\",{parentName:\"mrow\"},`i`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mn\",{parentName:\"mrow\"},`1`)),mdx(\"mn\",{parentName:\"munderover\"},`7`)),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mi\",{parentName:\"msub\"},`i`)),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"mo\",{parentName:\"mstyle\",\"stretchy\":\"false\"},`(`),mdx(\"msubsup\",{parentName:\"mstyle\"},mdx(\"mtext\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msubsup\"},`b`),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`i`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`))),mdx(\"mo\",{parentName:\"mstyle\"},`⋅`),mdx(\"mi\",{parentName:\"mstyle\",\"mathvariant\":\"bold-italic\"},`θ`),mdx(\"mo\",{parentName:\"mstyle\",\"stretchy\":\"false\"},`)`)),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)),mdx(\"mn\",{parentName:\"msup\"},`2`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nMSE(m,b) = \\\\frac{1}{7} \\\\cdot \\\\sum_{i=1}^7{(y_i-{\\\\color{#26a6ed}(\\\\textbf{x}^{(i)}_b \\\\cdot \\\\boldsymbol{\\\\theta})})}^2\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10903em\"}},`M`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`S`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`E`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.00744em\",\"verticalAlign\":\"-0.686em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen nulldelimiter\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"mfrac\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.314em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.23em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}})),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.677em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose nulldelimiter\"})),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"3.0787820000000004em\",\"verticalAlign\":\"-1.277669em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mop op-limits\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.8011130000000004em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.872331em\",\"marginLeft\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel mtight\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.050005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mop op-symbol large-op\"},`∑`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.3000050000000005em\",\"marginLeft\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.05em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`7`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.277669em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\",\"style\":{\"color\":\"#26a6ed\"}},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\",\"style\":{\"color\":\"#26a6ed\"}},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.0448em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.3986920000000005em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`b`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.2198em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen mtight\",\"style\":{\"color\":\"#26a6ed\"}},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`i`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose mtight\",\"style\":{\"color\":\"#26a6ed\"}},`)`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3013079999999999em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\",\"style\":{\"color\":\"#26a6ed\"}},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\",\"color\":\"#26a6ed\"}},`θ`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\",\"style\":{\"color\":\"#26a6ed\"}},`)`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.248808em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.4977em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`))))))))))))),mdx(\"p\",null,`Remember, the part of the equation that is colored in blue is equivalent to `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`x`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`f(x)`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`x`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`))))),`.\nOk, but what does this `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mrow\",{parentName:\"msup\"}),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`i`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`^{(i)}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8879999999999999em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8879999999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen mtight\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose mtight\"},`)`))))))))))))),` mean? Is this something like the “i-th power?”\nI’m glad that you ask!\nIf we write out our equation in more detail once again, we get:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`M`),mdx(\"mi\",{parentName:\"mrow\"},`S`),mdx(\"mi\",{parentName:\"mrow\"},`E`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"mi\",{parentName:\"mrow\"},`b`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},`1`),mdx(\"mn\",{parentName:\"mfrac\"},`7`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mn\",{parentName:\"mrow\"},`80`),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`(`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`)`)),mdx(\"mo\",{parentName:\"mstyle\"},`⋅`),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`(`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`)`))),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},`)`),mdx(\"mn\",{parentName:\"msup\"},`2`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mn\",{parentName:\"mrow\"},`170`),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`(`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`2`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`)`)),mdx(\"mo\",{parentName:\"mstyle\"},`⋅`),mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`(`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`)`))),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},`)`),mdx(\"mn\",{parentName:\"msup\"},`2`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},`.`),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},`.`),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},`.`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nMSE(m,b) =  \\\\frac{1}{7} \\\\cdot ((80-{\\\\color{#26a6ed}\\\\begin{pmatrix}1\\\\\\\\1\\\\end{pmatrix} \\\\cdot \\\\begin{pmatrix}\\\\theta_0\\\\\\\\\\\\theta_1\\\\end{pmatrix}})^2 + (170-{\\\\color{#26a6ed}\\\\begin{pmatrix}1\\\\\\\\2\\\\end{pmatrix} \\\\cdot \\\\begin{pmatrix}\\\\theta_0\\\\\\\\\\\\theta_1\\\\end{pmatrix}})^2 + ...)\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10903em\"}},`M`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`S`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`E`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.00744em\",\"verticalAlign\":\"-0.686em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen nulldelimiter\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"mfrac\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.314em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.23em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}})),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.677em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose nulldelimiter\"})),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`8`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.40003em\",\"verticalAlign\":\"-0.95003em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"minner\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`(`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`)`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\",\"style\":{\"color\":\"#26a6ed\"}},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`(`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`)`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8641079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.40003em\",\"verticalAlign\":\"-0.95003em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"minner\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`(`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},`2`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`)`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\",\"style\":{\"color\":\"#26a6ed\"}},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`(`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"color\":\"#26a6ed\",\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"color\":\"#26a6ed\"}},`)`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8641079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`2`)))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`.`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`.`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`.`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`)))))),mdx(\"p\",null,`As we see, `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msubsup\"},`b`),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`i`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}^{(i)}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.3461079999999999em\",\"verticalAlign\":\"-0.30130799999999996em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.0448em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.398692em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.2197999999999998em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen mtight\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose mtight\"},`)`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30130799999999996em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` is the `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`i`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.65952em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`i`))))),`-th sample in our `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\\\\textbf{x}_b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.59444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\"},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),`.\nSince we’ve already used the `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`))))),` as a subscript, we just put the `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`i`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.65952em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`i`))))),` at the top, as a superscript.\nWe write the `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`i`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.65952em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`i`))))),` in brackets to signal that this is not a power, but just an index-notation. I know\nthis looks a bit off when you first see it, but at some point, you will get used to it.\nNote that we could have just not used `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`b`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.69444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`))))),` as a subscript and put the `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`i`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.65952em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`i`))))),` at the bottom instead,\nbut this way we do can show all of the necessary information, even though it looks a bit bloated.`),mdx(\"p\",null,`And now, instead of taking the sum over every sample in our dataset, we instead calculate\nthe error for every sample at once by replacing our sum with the following equation:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`y`),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"mo\",{parentName:\"mstyle\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mtext\",{parentName:\"msub\",\"mathvariant\":\"bold\"},`x`),mdx(\"mi\",{parentName:\"msub\"},`b`)),mdx(\"mo\",{parentName:\"mstyle\"},`⋅`),mdx(\"mi\",{parentName:\"mstyle\",\"mathvariant\":\"bold-italic\"},`θ`),mdx(\"mo\",{parentName:\"mstyle\",\"stretchy\":\"false\"},`)`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\ny-{\\\\color{#26a6ed}(\\\\textbf{x}_b \\\\cdot \\\\boldsymbol{\\\\theta})}\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\",\"style\":{\"color\":\"#26a6ed\"}},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord text\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord textbf\",\"style\":{\"color\":\"#26a6ed\"}},`x`)),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.33610799999999996em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`b`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\",\"style\":{\"color\":\"#26a6ed\"}},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord boldsymbol\",\"style\":{\"marginRight\":\"0.03194em\",\"color\":\"#26a6ed\"}},`θ`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\",\"style\":{\"color\":\"#26a6ed\"}},`)`))))))),mdx(\"p\",null,`The part in blue is just the following:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`2`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`3`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`3`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`4`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`4`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`1`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`5`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`1`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`2`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`3`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`3`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`4`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`4`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`0`)),mdx(\"mo\",{parentName:\"mrow\"},`+`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`θ`),mdx(\"mn\",{parentName:\"msub\"},`1`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mn\",{parentName:\"mrow\"},`5`)))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`))))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\n\\\\left[\\\\begin{array}{c}\n\n    1 & 1 \\\\\\\\\n    1 & 2 \\\\\\\\\n    1 & 3 \\\\\\\\\n    1 & 3 \\\\\\\\\n    1 & 4 \\\\\\\\\n    1 & 4 \\\\\\\\\n    1 & 5 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n\\\\cdot\n\n\\\\left[\\\\begin{array}{c}\n\n    \\\\theta_0 \\\\\\\\\n    \\\\theta_1 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n=\n\n\\\\left[\\\\begin{array}{c}\n\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 1 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 2 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 3 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 3 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 4 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 4 \\\\\\\\\n    \\\\theta_0 + \\\\theta_1 \\\\cdot 5 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n=\n\n{\\\\color{#26a6ed}y_{pred}}\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.40003em\",\"verticalAlign\":\"-0.95003em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`[`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.45em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4099999999999997em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.9500000000000004em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size3\"},`]`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`+`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},`θ`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.30110799999999993em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.02778em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.716668em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"})))))))))))),mdx(\"p\",null,`Since this part represents the predicted prices of our function, we call it `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`))))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`{\\\\color{#26a6ed}y_{pred}}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.716668em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),`.`),mdx(\"p\",null,`Now we can calculate the squares of residuals by taking the dot product of `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`y`),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`))))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`y-{\\\\color{#26a6ed}y_{pred}}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.716668em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),` with itself:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`M`),mdx(\"mi\",{parentName:\"mrow\"},`S`),mdx(\"mi\",{parentName:\"mrow\"},`E`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mi\",{parentName:\"mrow\"},`m`),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},`,`),mdx(\"mi\",{parentName:\"mrow\"},`b`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mfrac\",{parentName:\"mrow\"},mdx(\"mn\",{parentName:\"mfrac\"},`1`),mdx(\"mn\",{parentName:\"mfrac\"},`7`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mi\",{parentName:\"msub\"},`i`)),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`)))),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},`)`),mdx(\"mi\",{parentName:\"msup\"},`T`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`(`),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mi\",{parentName:\"msub\"},`i`)),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`)))),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},`)`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\nMSE(m,b) = \\\\frac{1}{7} \\\\cdot  ( (y_i-{\\\\color{#26a6ed}y_{pred}})^T \\\\cdot (y_i-{\\\\color{#26a6ed}y_{pred}}) )\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10903em\"}},`M`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`S`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.05764em\"}},`E`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`m`),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},`,`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},`b`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"2.00744em\",\"verticalAlign\":\"-0.686em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen nulldelimiter\"}),mdx(\"span\",{parentName:\"span\",\"className\":\"mfrac\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"1.32144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.314em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`7`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.23em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"frac-line\",\"style\":{\"borderBottomWidth\":\"0.04em\"}})),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.677em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`1`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.686em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose nulldelimiter\"})),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.177439em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8913309999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.13889em\"}},`T`)))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},`(`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.31166399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`i`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.036108em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},`)`)))))),mdx(\"p\",null,`This is okay to do since taking the dot product of a `,mdx(\"em\",{parentName:\"p\"},`row vector`),` with a `,mdx(\"em\",{parentName:\"p\"},`column vector`),` yields us a scalar.\nIf we plug in the values for our concrete example and for our function `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`f`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`f`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8888799999999999em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.10764em\"}},`f`))))),`, we get:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},`y`),mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mstyle\",{parentName:\"mrow\",\"mathcolor\":\"#26a6ed\"},mdx(\"msub\",{parentName:\"mstyle\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`)))),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`20`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`50`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`80`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`40`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`40`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`30`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`200`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\ny - {\\\\color{#26a6ed}y_{pred}}\n\n=\n\n\\\\left[\\\\begin{array}{c}\n\n   20 \\\\\\\\\n   50 \\\\\\\\\n   -80 \\\\\\\\\n   40 \\\\\\\\\n   -40 \\\\\\\\\n   30 \\\\\\\\\n   200 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.7777700000000001em\",\"verticalAlign\":\"-0.19444em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.716668em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\",\"color\":\"#26a6ed\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\",\"style\":{\"color\":\"#26a6ed\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\",\"color\":\"#26a6ed\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"color\":\"#26a6ed\"}},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`8`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))))))))),mdx(\"p\",null,`and if we take the dot product of this (transposed) vector with itself, we get:`),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`20`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`50`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`80`)))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`40`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`40`)))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`30`))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`200`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\"},`⋅`),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`[`),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.15999999999999992em\",\"columnalign\":\"center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`20`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`50`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`80`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`40`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\"},`−`),mdx(\"mn\",{parentName:\"mrow\"},`40`))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`30`)))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},`200`))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},`]`)),mdx(\"mo\",{parentName:\"mrow\"},`=`),mdx(\"mn\",{parentName:\"mrow\"},`53400`)),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`\n\\\\left[\\\\begin{array}{c}\n\n   20 & 50 & -80 & 40 & -40 & 30 & 200 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n\\\\cdot\n\n\\\\left[\\\\begin{array}{c}\n\n   20 \\\\\\\\\n   50 \\\\\\\\\n   -80 \\\\\\\\\n   40 \\\\\\\\\n   -40 \\\\\\\\\n   30 \\\\\\\\\n   200 \\\\\\\\\n\n\\\\end{array}\\\\right]\n\n= 53400\n\n`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.20001em\",\"verticalAlign\":\"-0.35001em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size1\"},`[`)),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`8`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8500000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.01em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.35000000000000003em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing size1\"},`]`))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},`⋅`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222222222222222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"8.409030000000001em\",\"verticalAlign\":\"-3.9500599999999997em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎣`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎢`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎡`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mtable\"},mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"col-align-c\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.450000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.61em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.410000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.210000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`8`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.0100000000000002em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.8100000000000003em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`−`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.6100000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.5900000000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`2`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.95em\"}},mdx(\"span\",{parentName:\"span\"}))))),mdx(\"span\",{parentName:\"span\",\"className\":\"arraycolsep\",\"style\":{\"width\":\"0.5em\"}}))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizing mult\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"4.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"0.1500399999999994em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎦`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-0.9999600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-1.5959600000000007em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.191960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.787960000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.3839600000000005em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.9799600000000006em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-4.57596em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.17196em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-5.217950000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎥`))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-6.458970000000001em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"3.1550000000000002em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"delimsizinginner delim-size4\"},mdx(\"span\",{parentName:\"span\"},`⎤`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"3.9500599999999997em\"}},mdx(\"span\",{parentName:\"span\"}))))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},`=`),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2777777777777778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.64444em\",\"verticalAlign\":\"0em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`5`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`3`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`4`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},`0`)))))),mdx(\"p\",null,`which is in fact the same value that we got from our non-vectorized implementation of the MSE!`),mdx(\"p\",null,`If we translate our new definition into code, we get:`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`def mse_vectorized(y, y_predicted):\n    error = y-y_predicted # save y-y_pred in a variable so that we only have to compute it once\n    loss = 1/(y.size) * np.dot(error.T, error)\n    return loss\n`)),mdx(\"p\",null,`We now no longer need to pass in our entire dataset as well as a function to our MSE.\nInstead, we calculate `,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},`y`),mdx(\"mrow\",{parentName:\"msub\"},mdx(\"mi\",{parentName:\"mrow\"},`p`),mdx(\"mi\",{parentName:\"mrow\"},`r`),mdx(\"mi\",{parentName:\"mrow\"},`e`),mdx(\"mi\",{parentName:\"mrow\"},`d`)))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},`y_{pred}`)))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.716668em\",\"verticalAlign\":\"-0.286108em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.03588em\"}},`y`),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3361079999999999em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"-0.03588em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`p`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.02778em\"}},`r`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`e`),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},`d`))))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},`​`)),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.286108em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),` beforehand, like this: `,mdx(\"inlineCode\",{parentName:\"p\"},`y_predicted = f(x)`),`.\nThis makes our function a bit more lightweight.`),mdx(\"p\",null,`Great, now we have vectorized our MSE as well!`),mdx(\"h2\",{\"id\":\"speed-comparison\"},`Speed Comparison`),mdx(\"p\",null,`Ok, now let’s test both of our implementations on an exemplary dataset.\nBut for a real test, our small dataset is not going to cut it.\nSince our dataset does not really have to make sense, we can just use\nsome random numbers. Let’s create an array with roughly one million values\nand reshape it to contain individual arrays `,`[x,y]`,`. Then we can also save x and y\nseparately.`),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-python\"},`arranged_values = np.arange(1,100000,0.1).reshape(-1,2) #shape: (499995, 2)\narranged_values.shape\n\nx = arranged_values[:,0]\ny = arranged_values[:,1]\n`)),mdx(\"p\",null,`If we now time the executions of our non-vectorized and our vectorized MSE using the magic command\n`,mdx(\"inlineCode\",{parentName:\"p\"},`%%timeit`),` (you can learn more about those `,mdx(\"a\",{parentName:\"p\",\"href\":\"/development_environment/jupyter/#magic-commands\"},`here`),`),\nwe get this:`),mdx(\"p\",null,mdx(\"span\",{parentName:\"p\",\"className\":\"gatsby-resp-image-wrapper\",\"style\":{\"position\":\"relative\",\"display\":\"block\",\"marginLeft\":\"auto\",\"marginRight\":\"auto\",\"maxWidth\":\"999px\"}},`\n      `,mdx(\"span\",{parentName:\"span\",\"className\":\"gatsby-resp-image-background-image\",\"style\":{\"paddingBottom\":\"26.400000000000002%\",\"position\":\"relative\",\"bottom\":\"0\",\"left\":\"0\",\"backgroundImage\":\"url('data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAFCAYAAABFA8wzAAAACXBIWXMAABYlAAAWJQFJUiTwAAAAvklEQVQY011QWw7DIAzr/W9Yqdp+1hfQJ6WFMuTJqZi6IUU2hCSOC+dOrDbgOA7UD4WqqjAMA+Z5xjRNGMfxy9d1hfde/t4xhKt+33cUuB1jNLquk2iaBm3boq5rib7voZSCtRbOOUE24BAOJI8xojhjgg9vudjJiRIm77Ftmyhgo8wzsvGyLMJTSii8j9jcKbJfzxZlWYo6qtRaS0Feh3jn/28c+LPyvFxe0UOuR/9yZA+phshBRGOM5FlDDz9l8H/E/UZNbgAAAABJRU5ErkJggg==')\",\"backgroundSize\":\"cover\",\"display\":\"block\"}}),`\n  `,mdx(\"picture\",{parentName:\"span\"},`\n          `,mdx(\"source\",{parentName:\"picture\",\"srcSet\":[\"/static/68722276cbb1f849a98761021563f451/601b1/speedTestVectorization.webp 500w\",\"/static/68722276cbb1f849a98761021563f451/8d82a/speedTestVectorization.webp 999w\"],\"sizes\":\"(max-width: 999px) 100vw, 999px\",\"type\":\"image/webp\"}),`\n          `,mdx(\"source\",{parentName:\"picture\",\"srcSet\":[\"/static/68722276cbb1f849a98761021563f451/0eb09/speedTestVectorization.png 500w\",\"/static/68722276cbb1f849a98761021563f451/45359/speedTestVectorization.png 999w\"],\"sizes\":\"(max-width: 999px) 100vw, 999px\",\"type\":\"image/png\"}),`\n          `,mdx(\"img\",{parentName:\"picture\",\"className\":\"gatsby-resp-image-image\",\"src\":\"/static/68722276cbb1f849a98761021563f451/45359/speedTestVectorization.png\",\"alt\":\"Comparing non-vectorized and vectorized MSE\",\"title\":\"Comparing non-vectorized and vectorized MSE\",\"loading\":\"lazy\",\"decoding\":\"async\",\"style\":{\"width\":\"100%\",\"height\":\"100%\",\"margin\":\"0\",\"verticalAlign\":\"middle\",\"position\":\"absolute\",\"top\":\"0\",\"left\":\"0\"}}),`\n        `),`\n    `)),mdx(\"p\",null,`As you can see, the vectorized version outperformed the non-vectorized one by a landslide!`),mdx(\"h2\",{\"id\":\"further-reading\"},`Further reading`),mdx(\"p\",null,`If you are interested in applying vectorization to a linear regression problem, I highly recommend you read the post\n`,mdx(\"a\",{parentName:\"p\",\"href\":\"/machine_learning_models/linear_regression\"},`Linear Regression Explained, Step by Step`),`, where\nyou will learn everything you need to know about linear regression and\nuse vectorization in practice.`));};MDXContent.isMDXComponent=true;","excerpt":"Vectorization is one of the most useful techniques to make your machine learning code more efficient. In this post, you will learn everything you need to know to start using vectorization efficiently in your machine learning projects.","tableOfContents":{"items":[{"url":"#outline","title":"Outline"},{"url":"#the-main-idea","title":"The main idea"},{"url":"#vectorization","title":"Vectorization","items":[{"url":"#example-1-linear-function","title":"Example 1: Linear Function"},{"url":"#example-2-mse","title":"Example 2: MSE"}]},{"url":"#speed-comparison","title":"Speed Comparison"},{"url":"#further-reading","title":"Further reading"}]},"frontmatter":{"title":"Vectorization Explained, Step by Step","userDate":"6 March 2021","date":"2021-03-06T18:32:05.647Z","dateModified":"2021-03-06T18:32:05.647Z","changelog":[["2021-12-28T13:18:41.297Z","Fixed some issues in the math examples. Thanks to stumpeschach for pointing them out!"]],"tags":["Machine Learning Math"],"timeToRead":7,"excerpt":"Vectorization is one of the most useful techniques to make your machine learning code more efficient. In this post, you will learn everything you need to know to start using vectorization efficiently in your machine learning projects.","image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/48d3891f0b15a736311ac3dd1dd6eea6/87706/Vectorization_Thumbnail.png","srcSet":"/static/48d3891f0b15a736311ac3dd1dd6eea6/0dee1/Vectorization_Thumbnail.png 750w,\n/static/48d3891f0b15a736311ac3dd1dd6eea6/8beaa/Vectorization_Thumbnail.png 1080w,\n/static/48d3891f0b15a736311ac3dd1dd6eea6/87706/Vectorization_Thumbnail.png 1280w","sizes":"100vw"},"sources":[{"srcSet":"/static/48d3891f0b15a736311ac3dd1dd6eea6/a66aa/Vectorization_Thumbnail.webp 750w,\n/static/48d3891f0b15a736311ac3dd1dd6eea6/65dd5/Vectorization_Thumbnail.webp 1080w,\n/static/48d3891f0b15a736311ac3dd1dd6eea6/71d4d/Vectorization_Thumbnail.webp 1280w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5625}}},"author":[{"id":"Lari Giba","bio":"Hi! I'm Lari and I run this website. You can find more information in the \"About\"-tab.","avatar":{"children":[{"fluid":{"base64":"data:image/jpeg;base64,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","aspectRatio":0.7407407407407407,"src":"/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg","srcSet":"/static/9eab840ebd4087faf31c48702c92af36/b3d27/lari.jpg 40w,\n/static/9eab840ebd4087faf31c48702c92af36/5a735/lari.jpg 80w,\n/static/9eab840ebd4087faf31c48702c92af36/6e81a/lari.jpg 120w,\n/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg 800w,\n/static/9eab840ebd4087faf31c48702c92af36/75b83/lari.jpg 1520w","sizes":"(max-width: 800px) 100vw, 800px"}}]}}],"imgCaption":"Background image by Joel Filipe","imgSource":"https://unsplash.com/photos/ZMRMFULofus","uniquePostId":4,"tutorialId":null,"tutorialEntryIdx":null,"dependencies":[],"proficiencyLevel":"Beginner"}},"relatedPosts":{"totalCount":3,"edges":[{"node":{"id":"d331ba94-308d-5859-8c5e-3f7526c59b0a","excerpt":"Gradient descent is one of the most popular algorithms to train machine learning models. However, many of the popular machine learning models like lasso regression or support vector machines contain loss functions that are not differentiable. Because of this, regular gradient descent can not be used. One of the most commonly utilized techniques to circumvent this issue is to use subgradients instead of regular gradients. And in this article, you will learn how it's done.","frontmatter":{"title":"Subgradient Descent Explained, Step by Step","timeToRead":18,"date":"2021-07-27T19:49:21.503Z"},"fields":{"slug":"/machine_learning_math/subgradient_descent/"}}},{"node":{"id":"340d90b8-6082-5581-a444-263bddfa3711","excerpt":"Gradient descent is one of the most famous techniques in machine learning and used for training all sorts of neural networks. But gradient descent can not only be used to train neural networks, but many more machine learning models. In particular, gradient descent can be used to train a linear regression model! If you are curious as to how this is possible, or if you want to approach gradient descent with smaller steps and not jump straight to neural networks, this post is for you. You will learn how gradient descent works from an intuitive, visual, and mathematical standpoint and we will apply it to an exemplary dataset in Python.","frontmatter":{"title":"Gradient Descent for Linear Regression Explained, Step by Step","timeToRead":23,"date":"2021-04-03T19:07:33.321Z"},"fields":{"slug":"/machine_learning_math/gradient_descent_for_linear_regression/"}}},{"node":{"id":"8eab4786-e5fd-5dd7-bc3f-bec6b2d4b411","excerpt":"Vectorization is one of the most useful techniques to make your machine learning code more efficient. In this post, you will learn everything you need to know to start using vectorization efficiently in your machine learning projects.","frontmatter":{"title":"Vectorization Explained, Step by Step","timeToRead":7,"date":"2021-03-06T18:32:05.647Z"},"fields":{"slug":"/machine_learning_math/vectorization/"}}}]}},"pageContext":{"slug":"/machine_learning_math/vectorization/","prev":{"excerpt":"In this article you will learn what Jupyter and Jupyter Notebooks are, when to use them and how to use them. You will learn about the most common shortcuts, Jupyter Lab, as well as magic commands.","frontmatter":{"title":"The Only Post you Need to Start Using Jupyter","tags":["Development Environment"],"date":"2021-02-20T17:13:56.071Z","dateModified":"2021-03-22T10:07:41.913Z","changelog":[["2021-03-22T10:07:41.913Z","Added a paragraph about performance of Jupyter in the section 'Jupyter Integrations'."]],"draft":false,"excerpt":"In this article you will learn what Jupyter and Jupyter Notebooks are, when to use them and how to use them. You will learn about the most common shortcuts, Jupyter Lab, as well as magic commands.","uniquePostId":3,"timeToRead":11,"tutorialId":null,"tutorialEntryIdx":null,"dependencies":["Python"],"proficiencyLevel":"Beginner","image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/93f510c2f49caa940058b5d7c90f1871/87706/Jupyter_Thumbnail.png","srcSet":"/static/93f510c2f49caa940058b5d7c90f1871/0dee1/Jupyter_Thumbnail.png 750w,\n/static/93f510c2f49caa940058b5d7c90f1871/8beaa/Jupyter_Thumbnail.png 1080w,\n/static/93f510c2f49caa940058b5d7c90f1871/87706/Jupyter_Thumbnail.png 1280w","sizes":"100vw"},"sources":[{"srcSet":"/static/93f510c2f49caa940058b5d7c90f1871/a66aa/Jupyter_Thumbnail.webp 750w,\n/static/93f510c2f49caa940058b5d7c90f1871/65dd5/Jupyter_Thumbnail.webp 1080w,\n/static/93f510c2f49caa940058b5d7c90f1871/71d4d/Jupyter_Thumbnail.webp 1280w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5625}}},"author":[{"id":"Lari Giba","bio":"Hi! I'm Lari and I run this website. You can find more information in the \"About\"-tab.","avatar":{"children":[{"fluid":{"aspectRatio":0.7434944237918215,"base64":"data:image/jpeg;base64,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","sizes":"(max-width: 800px) 100vw, 800px","src":"/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg","srcSet":"/static/9eab840ebd4087faf31c48702c92af36/fd013/lari.jpg 200w,\n/static/9eab840ebd4087faf31c48702c92af36/25252/lari.jpg 400w,\n/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg 800w,\n/static/9eab840ebd4087faf31c48702c92af36/0ff54/lari.jpg 1200w,\n/static/9eab840ebd4087faf31c48702c92af36/75b83/lari.jpg 1520w"}}]}}]},"fields":{"layout":"post","slug":"/development_environment/jupyter/"}},"next":{"excerpt":"Linear regression is one of the most famous algorithms in statistics and machine learning. In this post you will learn how linear regression works on a fundamental level. You will also implement linear regression both from scratch as well as with the popular library scikit-learn in Python. You will learn when and how to best use linear regression in your machine learning projects. You do not need any knowledge prior to reading this article.","frontmatter":{"title":"Linear Regression Explained, Step by Step","tags":["Machine Learning Models"],"date":"2021-03-20T18:17:30.352Z","dateModified":"2021-03-22T10:07:41.913Z","changelog":[["2023-01-17T18:46:34.276Z","Fixed a small typo. Thanks Андрей for pointing this out!"],["2023-05-21T21:43:11.881Z","Fixed a small typo. Thanks John for pointing this out!"]],"draft":false,"excerpt":"Linear regression is one of the most famous algorithms in statistics and machine learning. In this post you will learn how linear regression works on a fundamental level. You will also implement linear regression both from scratch as well as with the popular library scikit-learn in Python. You will learn when and how to best use linear regression in your machine learning projects. You do not need any knowledge prior to reading this article.","uniquePostId":5,"timeToRead":23,"tutorialId":null,"tutorialEntryIdx":null,"dependencies":[],"proficiencyLevel":"Beginner","image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/c32af02d3a84ba704bed82458e50c3fb/87706/LinearRegression_Thumbnail.png","srcSet":"/static/c32af02d3a84ba704bed82458e50c3fb/0dee1/LinearRegression_Thumbnail.png 750w,\n/static/c32af02d3a84ba704bed82458e50c3fb/8beaa/LinearRegression_Thumbnail.png 1080w,\n/static/c32af02d3a84ba704bed82458e50c3fb/87706/LinearRegression_Thumbnail.png 1280w","sizes":"100vw"},"sources":[{"srcSet":"/static/c32af02d3a84ba704bed82458e50c3fb/a66aa/LinearRegression_Thumbnail.webp 750w,\n/static/c32af02d3a84ba704bed82458e50c3fb/65dd5/LinearRegression_Thumbnail.webp 1080w,\n/static/c32af02d3a84ba704bed82458e50c3fb/71d4d/LinearRegression_Thumbnail.webp 1280w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5625}}},"author":[{"id":"Lari Giba","bio":"Hi! I'm Lari and I run this website. You can find more information in the \"About\"-tab.","avatar":{"children":[{"fluid":{"aspectRatio":0.7434944237918215,"base64":"data:image/jpeg;base64,/9j/2wBDABALDA4MChAODQ4SERATGCgaGBYWGDEjJR0oOjM9PDkzODdASFxOQERXRTc4UG1RV19iZ2hnPk1xeXBkeFxlZ2P/2wBDARESEhgVGC8aGi9jQjhCY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2NjY2P/wgARCAAbABQDASIAAhEBAxEB/8QAGAAAAwEBAAAAAAAAAAAAAAAAAAMEAgH/xAAWAQEBAQAAAAAAAAAAAAAAAAABAgP/2gAMAwEAAhADEAAAAeDIop5sDKn0aCSgzv8A/8QAHhABAAICAQUAAAAAAAAAAAAAAQACAxEEEyEiMjP/2gAIAQEAAQUC3ZAE6ZGW8HtMw1M2O+tTlfI9Z//EABURAQEAAAAAAAAAAAAAAAAAABAR/9oACAEDAQE/AYU//8QAFxEBAAMAAAAAAAAAAAAAAAAAABAREv/aAAgBAgEBPwFbMf/EABwQAAIBBQEAAAAAAAAAAAAAAAABMQIQESFxkf/aAAgBAQAGPwLfpJvBhuRVIgXRNwpsxct//8QAHRABAAMAAgMBAAAAAAAAAAAAAQARIUFRMWGBcf/aAAgBAQABPyG78vaUcDUeYJrJrCUcHAhzK6pYRVinqYAdKT7Y0/Yi+8UGE//aAAwDAQACAAMAAAAQsO1C/8QAGBEBAQEBAQAAAAAAAAAAAAAAAREAECH/2gAIAQMBAT8QPF4uTBS7/8QAGBEBAQEBAQAAAAAAAAAAAAAAAQARIVH/2gAIAQIBAT8QO2vYEgOZf//EAB8QAQEAAgIBBQAAAAAAAAAAAAERACExQVFhcZGxwf/aAAgBAQABPxDlRMDhDreGljrJ8uK5DiW/WJE4rXSx7wgAOy8sEKuzsxERAF59XxirS0k0dz03g4T3iY5jH9MRu2FvejEgEAAM/9k=","sizes":"(max-width: 800px) 100vw, 800px","src":"/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg","srcSet":"/static/9eab840ebd4087faf31c48702c92af36/fd013/lari.jpg 200w,\n/static/9eab840ebd4087faf31c48702c92af36/25252/lari.jpg 400w,\n/static/9eab840ebd4087faf31c48702c92af36/2f1b1/lari.jpg 800w,\n/static/9eab840ebd4087faf31c48702c92af36/0ff54/lari.jpg 1200w,\n/static/9eab840ebd4087faf31c48702c92af36/75b83/lari.jpg 1520w"}}]}}]},"fields":{"layout":"post","slug":"/machine_learning_models/linear_regression/"}},"primaryTag":"Machine Learning Math"}},"staticQueryHashes":["13077718","1316742457","1394499604","2907728042","3762220167"]}