{"id":229,"date":"2019-09-10T18:13:07","date_gmt":"2019-09-10T18:13:07","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=229"},"modified":"2020-06-18T17:42:40","modified_gmt":"2020-06-18T17:42:40","slug":"matrix-algebra","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/linear-algebra\/matrix-algebra\/","title":{"rendered":"Matrix Algebra"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Matrix Algebra &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nWe review here some of the basic definitions and elementary algebraic\noperations on matrices.\n\n<\/p>\n\n\n\n<p>\nThere are many applications as well as much interesting theory\nrevolving around these concepts, which we encourage you to explore\nafter reviewing this tutorial.\n\n<\/p>\n\n\n\n<p> A <b>matrix<\/b> is simply a retangular array of numbers.  For example, $$ A = \\left[ \\begin{array}{llcl} a_{11} &amp; a_{12} &amp; \\cdots &amp; a_{1n}\\\\ a_{21} &amp; a_{22} &amp; \\cdots &amp; a_{2n}\\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots\\\\ a_{m1} &amp; a_{m2} &amp; \\cdots &amp; a_{mn}\\\\ \\end{array} \\right] $$ is a $m\\times n$ matrix ($m$ rows, $n$ columns), where the entry in the $i^{th}$ row and $j^{th}$ column is $a_{ij}$.  We often write $A = [a_{ij}]$.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Some Terminology<\/h4>\n\n\n\n<p>\n\nFor an $n \\times n$ <b>square<\/b> matrix A, the elements\n$a_{11},a_{22},\\ldots,a_{nn}$ form the <b>main diagonal<\/b> of the\nmatrix.  The sum $\\sum\\limits_{k=1}^{n}a_{kk}$ of the elements on the main\ndiagonal of $A$ is called the <b>trace<\/b> of $A$.\n\n<\/p>\n\n\n\n<p>\nThe matrix $A^{T}=[a_{ji}]$ formed by interchanging the rows and\ncolumns of $A$ is called the <b>transpose<\/b> of $A$.  If $A^{T}=A$,\nthe matrix $A$ is <b>symmetric<\/b>.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t Let $\n\t\tB= \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]$.\n\t\tThe trace of $B$ is $6 + (-6) =0$.\n\t\t<\/p>\n\n\n\n<p>\n\t\tThe transpose of $B$ is $B^{T} =\n\t\t\\left[ \\begin{array}{ll}\n\t\t\t6 &amp; -4\\\\\n\t\t\t9 &amp; -6\\\\\n\t\t\\end{array} \\right]$.\n\t\t<br><br>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Addition and Subtraction of Matrices<\/h4>\n\n\n\n<p>\n\nTo <b>add<\/b> or <b>subtract<\/b> two matrices of the same size, simply\nadd or subtract corresponding entries.  That is, if $B=[b_{ij}]$ and\n$C=[c_{ij}]$, \n$$\nB + C = [b_{ij} + c_{ij}] {\\small\\textrm{ and }} B-C = [b_{ij} &#8211; c_{ij}]. \n$$\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t For $B = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]$ and $C = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right]$,\n\n\t\t$$\n\t\tB + C = \\left[ \n\t\t\\begin{array}{cc}\n\t\t\t6 +1 &amp; 9+2 \\\\\n\t\t\t-4+(-1) &amp; -6+0\\\\\n\t\t\\end{array}\n\t\t\\right] = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t7 &amp; 11 \\\\\n\t\t\t-5 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]\n\t\t$$\n\n\t\t$$\n\t\tB-C =  \\left[ \n\t\t\\begin{array}{cc}\n\t\t\t6 -1 &amp; 9-2 \\\\\n\t\t\t-4-(-1) &amp; -6-0\\\\\n\t\t\\end{array}\n\t\t\\right] = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t5 &amp; 7 \\\\\n\t\t\t-3 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right].\n\t\t$$\n<\/p>\n\n\n\n<p>\nThe $m\\times n$ <b>zero matrix<\/b>, <b>0<\/b>, for which every entry is 0, has\nthe property that for any $m\\times n$ matrix $A$, \n$$\nA+\\mathbf{0} = A.\n$$\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Scalar Multiplication<\/h4>\n\n\n\n<p>\n\nTo multiply a matrix $A$ by a number $c$ (a &#8220;scalar&#8221;), multiply each\nentry of $A$ by $c$.  That is,\n$$\ncA=[ca_{ij}].\n$$\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t Using the matrix \n\t\t$B= \\left[ \\begin{array}{rr} \n\t\t\t6 &amp; 9\\\\\n\t\t\t-4 &amp;-6\\\\\n\t\t\\end{array} \\right]$ from the previous example,\n\t\t$$\n\t\t3B = 3\\left[ \\begin{array}{rr} \n\t\t\t6 &amp; 9\\\\\n\t\t\t-4 &amp;-6\\\\\n\t\t\\end{array}\\right] = \\left[ \\begin{array}{rr} \n\t\t\t18 &amp; 27\\\\\n\t\t\t-12 &amp;-18\\\\\n\t\t\\end{array}\\right].\n\t\t$$\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Matrix Multiplication<\/h4>\n\n\n\n<p>\n\nLet $X$ be and $m \\times n$ matrix and $Y$ be an $n \\times p$ matrix.\nThen the <b>product<\/b> $XY$ is the $m\\times p$ matrix whose\n$(i,j)^{th}$ entry is given by \n$$\n\\sum^{n}_{k=1} x_{ik}y_{kj}.\n$$\n\n<\/p>\n\n\n\n<center>\n<br><br>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> The product $XY$ is only defined if the number of columns of $X$\n\t\tis the same as the number of rows of $Y$.\n\t<br><br>\n\t<\/li><li> $XY$ and $YX$ may very well not both be defined.  If they both\n\t\tdo exist, they are not necessarily equal and in fact might not even be\n\t\tof the same size.\n<\/li><\/ul>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t For the matrices $B = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]$ and $C = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right]$,\n\t\t$$\n\t\t{ BC = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right] \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right] = \\left[ \\begin{array}{cc}\n\t\t\t(6)(1)+(9)(-1) &amp; (6)(2)+(9)(0)\\\\\n\t\t\t(-4)(1)+(-6)(-1) &amp; (-4)(2) + (-6)(0)\\\\\n\t\t\\end{array} \\right] = \\left[ \\begin{array}{cc}\n\t\t\t-3 &amp; 12 \\\\\n\t\t\t2 &amp; -8\\\\\n\t\t\\end{array} \\right]}\n\t\t$$\n\t\twhile\n\t\t$$\n\t\t{ CB = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right] \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right] = \\left[ \\begin{array}{cc}\n\t\t\t(1)(6) + (2)(-4) &amp; (1)(9) + (2)(-6)\\\\\n\t\t\t(-1)(6) + (0)(-4) &amp; (-1)(9) + (0)(-6)\\\\\n\t\t\\end{array} \\right] = \\left[ \\begin{array}{cc}\n\t\t\t-2 &amp; -3 \\\\\n\t\t\t-6 &amp; -9\\\\\n\t\t\\end{array} \\right]}.\n\t\t$$\n<\/p>\n\n\n\n<p>\nThe $n\\times n$ matrix having all main diagonal entries equal to 1 and\nall other entries equal to 0 is called the <b>identity<\/b> matrix\n$I$.  For example, the $3\\times 3$ matrix is $\\left[ \n\\begin{array}{lll}\n\t1 &amp; 0 &amp; 0\\\\\n\t0 &amp; 1 &amp; 0\\\\\n\t0 &amp; 0 &amp; 1\\\\\n\\end{array} \\right]$.  The $n \\times n$ identity matrix has the\nproperty that if $A$ is any $n \\times n$ matrix, \n$$\nAI = IA = A.\n$$\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Inverse of a Matrix<\/h4>\n\n\n\n<p>\n\nStart with an $n \\times n$ matrix $X$.  Suppose the $n \\times n$\nmatrix $Y$ has the property that \n$$\nXY = YX = I.\n$$\nThen $Y$ is called the <b>inverse<\/b> of $X$ and is denoted $X^{-1}$.\n\n<\/p>\n\n\n\n<center>\n<br>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> Only square matrices $X$ can have inverses. If X is <i>not<\/i> square,  then for any Y the product XY will not be the same size matrix as the product YX , assuming both products exist. <br><br> <\/li><li> Not every square matrix has an inverse.  If an inverse exists, it is unique. <br><br> <\/li><li> If a matrix has an inverse, the matrix is said to be <b>invertible<\/b>. <\/li><\/ul>\n\n\n\n<p>\nThe inverse of a $2 \\times 2$ matrix is simple to calculate:\n$$\n{\\small\\textrm{If }} A=\\left[ \\begin{array}{rr}\n\ta &amp; b\\\\\n\tc &amp; d\\\\\n\\end{array} \\right], {\\small\\textrm{ then }} A^{-1} = \\frac{1}{ad-bc} \\left[ \n\\begin{array}{rr}\n\td &amp; -b\\\\\n\t-c &amp; a\\\\\n\\end{array} \\right].\n$$\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t The inverse of $ C = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right]$ is\n\t\t$$\n\t\tC^{-1} = \\frac{1}{(1)(0)-(2)(-1)}\\left[ \\begin{array}{rr}\n\t\t\t0 &amp; -2\\\\\n\t\t\t1 &amp; 1\\\\\n\t\t\\end{array} \\right] = \\frac{1}{2} \\left[ \\begin{array}{rr}\n\t\t\t0 &amp; -2\\\\\n\t\t\t1 &amp; 1\\\\\n\t\t\\end{array} \\right] = \\left[ \\begin{array}{rr}\n\t\t\t0 &amp; -1\\\\\n\t\t\t1\/2 &amp; 1\/2\\\\\n\t\t\\end{array} \\right].\n\t\t$$\n\t\tNote that $CC^{-1} = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right]\\left[ \\begin{array}{rr}\n\t\t\t0 &amp; -1\\\\\n\t\t\t1\/2 &amp; 1\/2\\\\\n\t\t\\end{array} \\right] = \\left[ \\begin{array}{rr}\n\t\t\t1 &amp; 0\\\\\n\t\t\t0 &amp; 1\\\\\n\t\t\\end{array} \\right]$\n\n\t\t<\/p>\n\n\n\n<p>\n\t\t$\\quad$\n\t\tand $C^{-1}C = \\left[ \\begin{array}{rr}\n\t\t\t0 &amp; -1\\\\\n\t\t\t1\/2 &amp; 1\/2\\\\\n\t\t\\end{array} \\right]\\left[ \n\t\t\\begin{array}{rr}\n\t\t\t1 &amp; 2 \\\\\n\t\t\t-1 &amp; 0\\\\\n\t\t\\end{array}\n\t\t\\right] = \\left[ \\begin{array}{rr}\n\t\t\t1 &amp; 0\\\\\n\t\t\t0 &amp; 1\\\\\n\t\t\\end{array} \\right].$\n\n\t\t<\/p>\n\n\n\n<p>\n\t\tMatrix $B = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]$ does not have an inverse.\n\t<br><br><br>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Determinant of a Matrix<\/h4>\n\n\n\n<p>\n\nHow did we know that $B = \\left[ \n\\begin{array}{rr}\n\t6 &amp; 9 \\\\\n\t-4 &amp; -6\\\\\n\\end{array}\n\\right]$ does not have an inverse?\n\n<\/p>\n\n\n\n<p>\nThe <b>determinant<\/b> of $A$, $\\det A$, is a number with the property\nthat $A$ is invertible if and only if $\\det A \\not= 0$.\n\n<\/p>\n\n\n\n<p>\nFor a $2 \\times 2$ matrix $A=\\left[ \\begin{array}{rr}\n\ta &amp; b\\\\\n\tc &amp; d\\\\\n\\end{array} \\right]$, $\\det A = ad -bc$.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\t\t For $B = \\left[ \n\t\t\\begin{array}{rr}\n\t\t\t6 &amp; 9 \\\\\n\t\t\t-4 &amp; -6\\\\\n\t\t\\end{array}\n\t\t\\right]$, $\\det B = (6)(-6)- (9)(-4)=-36+36=0$, so $B$ is not\n\t\tinvertible.  That is, $B$ does not have an inverse.\n<\/p>\n\n\n\n<p> For a $3 \\times 3$, or larger, matrix $A$, things are a little more complicated:  <\/p>\n\n\n\n<p>\n\t\tDenote by $M_{ij}(A)$ the determinant of the matrix formed by\n\t\tdeleting row $i$ and column $j$ for $A$.<\/p>\n\n\n\n<p>\n\t\tDefine $c_{ij}(A) = (-1)^{i+j}M_{ij}(A)$ to be the <b>$(i,j)$ cofactor<\/b> of $A$.<\/p>\n\n\n\n<p>\n\t\tThen we can compute $\\det A$ by the <b>Laplace Expansion<\/b> along any row or column of $A$:\n\t\t<\/p>\n\n\n\n<p>\n\t\t\t\t <b>Along row $i$:<\/b>\n\t\t\t\t$$\n\t\t\t\t\\det A = a_{i1}c_{i1}(A) + a_{i2}c_{i2}(A) + \\ldots + a_{in}c_{in}(A).\n\t\t\t\t$$\n\t\t\t\t\n\t\t\t\t<\/p>\n\n\n\n<p>\n\t\t\t\t <b>Along column $j$:<\/b>\n\t\t\t\t$$\n\t\t\t\t\\det A = a_{1j}c_{1j}(A) + a_{2j}c_{2j}(A)+ \\ldots + a_{nj}c_{nj}(A).\n\t\t\t\t$$\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p> Let $A = \\left[ \\begin{array}{rrr} 1 &amp; -1 &amp; 3\\\\ 1 &amp; 0 &amp; -1\\\\ 2 &amp; 1 &amp; 6\\\\ \\end{array} \\right].$ <\/p>\n\n\n\n<p>\n\tAlong the first row, \n\n\t<\/p>\n\n\n\n<p>\n\t$\\begin{array}{rcl}\n\t\t\\det A &amp; = &amp; (1) \\left[ (0)(6) &#8211; (-1)(1) \\right] &#8211; (-1)\\left\n\t\t[ (1)(6)-(-1)(2) \\right] + 3 \\left[ (1)(1)-(0)(2) \\right] \\\\\n\t\t &amp; = &amp; (1)(1) + (1)(8)+(3)(1)\\\\\n\t\t &amp; = &amp; 12.  \n\t\\end{array}$\n\n\t<\/p>\n\n\n\n<p>\n\tComputing $\\det A$ along the second column instead,\n\n\t<\/p>\n\n\n\n<p>\n\t$\\begin{array}{rcl}\n\t\t\\det A &amp; = &amp; -(-1) \\left[ (1)(6) &#8211; (-1)(2) \\right] + ( 0)\\left\n\t\t[ (1)(6)-(3)(2) \\right] &#8211; 1 \\left[ (1)(-1)-(3)(1) \\right] \\\\\n\t\t&amp; = &amp; (1)(8)+(0)(0)-(1)(-4)\\\\\n\t\t&amp; = &amp; 12  {\\small\\textrm{ as expected.}}\n\t\\end{array}$\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p>\n\nLet $A = [a_{ij}] {\\small\\textrm{ and }} B=[b_{ij}] $. \n\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> Transpose $A^{T}$  of $A$: <p> $$ A^{T}=[a_{ji}]. $$ <br><br> <\/p><\/li><li> Trace of $A$: <p> $$ \\sum_{k=1}^{n} a_{kk} {\\small\\textrm{ (for an }}n\\times n {\\small\\textrm{ matrix }} A). $$ <br><br> <\/p><\/li><li> Identity Matrix $I$: <p> the $n \\times n$ matrix with 1&#8217;s on the main digonal and 0&#8217;s elsewhere. <br><br> <\/p><\/li><li> <b>$A+B$ and $A-B$:<\/b> <p> $$ A+B = [a_{ij}+ b_{ij}] \\phantom{.} $$ $$ A-B = [a_{ij} -b_{ij}]. $$ <br><br> <\/p><\/li><li> <b>Scalar Multiplication:<\/b> <p> $$ cA = [ca_{ij}]. $$ <br><br> <\/p><\/li><li> <b>Matrix Product $AB$:<\/b> <p> $(i,j)^{th}$ entry is $\\sum\\limits_{k=1}^{n} a_{ik}b_{kj}$, for an $m \\times n$ matrix $A$ and an $n \\times p$ matrix $B$. <br><br> <\/p><\/li><li> <b>Inverse $A^{-1}$ of $A$:<\/b> <p> $A^{-1}$ satisfies $AA^{-1} = A^{-1}A = I$.  If $A = \\left[ \\begin{array}{ll} a &amp; b\\\\ c &amp; d\\\\ \\end{array}\\right]$,   then $A^{-1} = \\frac{1}{ad-bc} \\left[ \\begin{array}{rr} d &amp; -b \\\\ -c &amp; a \\\\ \\end{array}\\right].$ <br><br> <\/p><\/li><li> <b>Determinant $\\det A$:<\/b> <p> If $A = \\left[ \\begin{array}{ll} a &amp; b\\\\ c &amp; d\\\\ \\end{array}\\right]$, $\\det A = ad-bc$.  In general,  $\\qquad$ along row $i$:   $\\qquad\\qquad$ $\\det A = a_{i1}c_{i1}(A) + a_{i2}c_{i2}(A) + \\ldots + a_{in}c_{in}(A)$.  $\\qquad$ along column $j$:   $\\qquad\\qquad$  $\\det A = a_{1j}c_{1j}(A) + a_{2j}C_{2j}(A)+ \\ldots + a_{nj}c_{nj}(A)$. <\/p><\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p> [<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ2810\/\">I&#8217;m ready to take the quiz.<\/a>] [<a href=\"#top\">I need to review more.<\/a>]<br> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Matrix Algebra &#8211; HMC Calculus Tutorial We review here some of the basic definitions and elementary algebraic operations on matrices. There are many applications as well as much interesting theory revolving around these concepts, which we encourage you to explore after reviewing this tutorial. A matrix is simply a retangular array of numbers. For example,&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":61,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-229","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/229","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=229"}],"version-history":[{"count":7,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/229\/revisions"}],"predecessor-version":[{"id":1243,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/229\/revisions\/1243"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/61"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=229"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}