{"id":221,"date":"2019-09-10T17:01:06","date_gmt":"2019-09-10T17:01:06","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=221"},"modified":"2020-12-18T21:21:16","modified_gmt":"2020-12-18T21:21:16","slug":"change-of-basis","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/linear-algebra\/change-of-basis\/","title":{"rendered":"Change of Basis"},"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>Change of Basis &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p> Let $V$ be a vector space and let $S = \\{{\\bf v_1,v_2, \\ldots, v_n}\\}$ be a set of vectors in $V$.  Recall that $S$ forms a <strong>basis<\/strong> for $V$ if the following two conditions hold:    Let $V$ be a vector space and let $S = \\{{\\bf v_1,v_2, \\ldots, v_n}\\}$ be a set of vectors in $V$.  Recall that $S$ forms a <strong>basis<\/strong> for $V$ if the following two conditions hold:  <\/p>\n\n\n<style>.kt-accordion-id_e9d558-26 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_e9d558-26 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_e9d558-26 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_e9d558-26 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_e9d558-26 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_e9d558-26 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_e9d558-26 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_e9d558-26:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_e9d558-26 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_e9d558-26 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_e9d558-26 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_2befe1-27\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">$S$ is linearly independent<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\nLet $S = \\lbrace {\\bf v}_1 , {\\bf v}_2 , \\dots , {\\bf v}_n \\rbrace$ be a non-empty set of vectors. If $k_1 {\\bf v}_1 + k_2 {\\bf v}_2 + \\dots + k_n {\\bf v}_n = 0$ only when $k_1 , k_2 , \\dots , k_n = 0$, then $S$ is <strong>linearly independent<\/strong>. \n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_31c3d9-de\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">$S$ spans $V$<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\n Let $V$ be a vector space and let $\\lbrace {\\bf v}_1 , {\\bf v}_2 , \\dots , {\\bf v}_n \\rbrace$ be a set of elements in $V$. If every vector in $V$ can be expressed as a linear combination of ${\\bf v}_1 , {\\bf v}_2 , \\dots , {\\bf v}_n$ then $\\lbrace {\\bf v}_1 , {\\bf v}_2 , \\dots , {\\bf v}_n \\rbrace$ <strong>spans<\/strong> $V$. \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p> If $S = \\{{\\bf v_1,v_2, \\ldots, v_n}\\}$ is a basis for $V$, then every vector ${\\bf v} \\in V$ can be expressed <i>uniquely<\/i> as a linear combination of ${\\bf v_1,v_2, \\ldots, v_n}$: $$ {\\bf v} = c_1{\\bf v_1} + c_2{\\bf v_2} + \\cdots + c_n{\\bf v_n}. $$ Think of $\\left[\\begin{array}{c} c_1\\\\ c_2\\\\ \\vdots \\\\ c_n \\end{array}\\right]$ as the <b>coordinates<\/b> of ${\\bf v}$ relative to the basis $S$.  If $V$ has <b>dimension<\/b>, which is the number of vectors needed to form a basis.  $n$, then every set of $n$ linearly independent vectors in $V$ forms a basis for $V$.  In every application, we have a choice as to what basis we use.  In this tutorial, we will desribe the transformation of coordinates of vectors under a change of basis.<\/p>\n\n\n\n<p>We will focus on vectors in $R^2$, although all of this generalizes to $R^n$.  The standard basis in $R^2$ is $ \\left\\{\\left[{1 \\atop 0} \\right],\\left[{0 \\atop 1}\\right]\\right\\}$.  We specify other bases with reference to this rectangular coordinate system. <\/p>\n\n\n\n<p>\nLet $B=\\{{\\bf u,w}\\}$ and $B&#8217;=\\{{\\bf u&#8217;,w&#8217;}\\}$ be two bases for\n$R^2$.  For a vector ${\\bf v} \\in V$, given its coordinates $[{\\bf\nv}]_B$ in basis $B$ we would like to be able to express ${\\bf v}$ in\ntems of its coordinates $[{\\bf v}]_{B&#8217;}$ in basis $B&#8217;$, and vice versa.\n\n<\/p>\n\n\n\n<p>\nSuppose the basis vectors ${\\bf u&#8217;}$ and ${\\bf w&#8217;}$ for $B&#8217;$ have the\nfollowing coordinates relative to the basis $B$:\n\n<\/p>\n\n\n\n<p>\n\t\t\\begin{eqnarray*}\n\t\t\t~[{\\bf u&#8217;}]_B &amp; = &amp; \\left[\\begin{array}{c} a \\\\ b \\end{array}\\right] \\qquad \\\\\n\t\t\t~[{\\bf w&#8217;}]_B &amp; = &amp; \\left[\\begin{array}{c} c \\\\ d \\end{array}\\right]. \\qquad \n\t\t\\end{eqnarray*} \n<\/p>\n\n\n\n<p>\n\t\tThis means that \n\t\t\\begin{eqnarray*}\n\t\t\t{\\bf u&#8217;} &amp; = &amp; a{\\bf u} + b{\\bf w} \\\\\n\t\t\t{\\bf w&#8217;} &amp; = &amp; c{\\bf u} + d{\\bf w} \n\t\t\\end{eqnarray*}\n<\/p>\n\n\n\n<p> The <b>change of coordinates matrix<\/b> from $B&#8217;$ to $B$ $$ P = \\left[\\begin{array}{cc} a &amp; c \\\\ b &amp; d \\\\ \\end{array} \\right] $$ governs the change of coordinates of ${\\bf v} \\in V$ under the change of basis from $B&#8217;$ to $B$. $$ [{\\bf v}]_B = P[{\\bf v}]_{B&#8217;} = \\left[\\begin{array}{cc} a &amp; c \\\\ b &amp; d \\\\ \\end{array} \\right][{\\bf v}]_{B&#8217;}. $$ That is, if we know the coordinates of ${\\bf v}$ relative to the basis $B&#8217;$, multiplying this vector by the change of coordinates matrix gives us the coordinates of ${\\bf v}$ relative to the basis $B$. <\/p>\n\n\n<style>.kt-accordion-id_017ee6-de .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_017ee6-de .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_017ee6-de > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > 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.kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_017ee6-de .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_017ee6-de > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_017ee6-de:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_017ee6-de .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_017ee6-de .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_017ee6-de kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_8e0a31-2d\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Why?<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\nSuppose vector ${\\bf v}$ has coordinates $\\left[{x&#8217; \\atop y&#8217;}\\right]_{B&#8217;}$ relative to the basis $B&#8217; = \\{ {\\bf u&#8217;,w&#8217;} \\}$.  This means that  $$ {\\bf v} = x'{\\bf u&#8217;} + y'{\\bf w&#8217;}. $$ Substituting ${\\bf u&#8217;} = a{\\bf u} + b{\\bf w}$ and ${\\bf w&#8217;} = c{\\bf u} + d{\\bf w}$ into this, \\begin{eqnarray*} {\\bf v} &amp; = &amp; x'(a{\\bf u} + b{\\bf w}) + y'(c{\\bf u} + d{\\bf w}) \\\\  &amp; = &amp; (ax&#8217; + cy&#8217;){\\bf u} + (bx&#8217;+dy&#8217;){\\bf w}.  \\end{eqnarray*} That is, \\begin{eqnarray*} [{\\bf v}]_{B} &amp; = &amp; \\left[\\begin{array}{c} ax&#8217; + cy&#8217; \\\\ bx&#8217;+dy&#8217; \\end{array}\\right] \\\\  &amp; = &amp; \\left[\\begin{array}{cc} a &amp; c \\\\ b &amp; d \\end{array} \\right]\\left[\\begin{array}{c} x&#8217; \\\\ y&#8217; \\end{array}\\right] \\\\  &amp; = &amp; \\left[\\begin{array}{cc} a &amp; c \\\\ b &amp; d \\end{array} \\right][{\\bf v}]_{B&#8217;}. \\end{eqnarray*}  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p> The transition matrix $P$ is <b>invertible<\/b>.  <\/p>\n\n\n<style>.kt-accordion-id_dea61b-09 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:8px;}.kt-accordion-id_dea61b-09 .kt-accordion-panel-inner{padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_dea61b-09 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{padding-top:var(--global-kb-spacing-xxs, 0.5rem);padding-right:var(--global-kb-spacing-xs, 1rem);padding-bottom:var(--global-kb-spacing-xxs, 0.5rem);padding-left:var(--global-kb-spacing-xs, 1rem);}@media all and (max-width: 767px){.kt-accordion-id_dea61b-09 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_dea61b-09 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:8px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_dea61b-09 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_d31a5e-00\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Definition of Invertible<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n<!------------------------>\n\nIf a square matrix ${\\bf A}$ has an inverse ${\\bf A}^{-1}$, it is said to be <b>invertible<\/b>.\n\n<\/p>\n\n\n\n<p>\nIt can be shown that ${\\bf A}$ is invertible if and only if $\\det{\\bf A} \\neq {\\bf 0}$.\n\n<\/p>\n\n\n\n<p>\nFor a $2 \\times 2$ matrix \n\t${\\bf A} = \\left[ \\begin{array}{cc}\n\t\ta\t&amp;\tb \\\\\n\t\tc\t&amp;\td\n\t\\end{array} \\right], \\,\n\t{\\bf A}^{-1} = \\frac{1}{ad-bc} \\left[ \\begin{array}{cc}\n\t\td\t&amp;\t-b \\\\\n\t\t-c\t&amp;\ta\n\t\\end{array} \\right].$\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>In fact, if $P$ is the change of coordinates matrix from $B&#8217;$ to $B$, the $P^{-1}$ is the change of coordinates matrix from $B$ to $B&#8217;$: $$ [{\\bf v}]_{B&#8217;} = P^{-1}[{\\bf v}]_B $$<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/2bases.gif?resize=300%2C300&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-335\"\/><\/figure><\/div>\n\n\n\n<p>\nLet $B = \\left\\{\\left[{1 \\atop 0} \\right],\\left[{0 \\atop\n1}\\right]\\right\\}$ and $B&#8217; = \\left\\{\\left[{3 \\atop 1} \\right],\\left[{-2 \\atop\n1}\\right]\\right\\}$.  The change of basis matrix form $B&#8217;$ to $B$ is\n$$\nP = \\left[\\begin{array}{cc}\n\t3 &amp; -2 \\\\\n\t1 &amp; 1\n\\end{array}\\right].\n$$\nThe vector ${\\bf v}$ with coordinates $[{\\bf v}]_{B&#8217;} = \\left[ {2 \\atop 1}\n\\right]$ relative to the basis $B&#8217;$ has coordinates\n$$\n[{\\bf v}]_B = \\left[ \\begin{array}{cc}\n\t3 &amp; -2 \\\\\n\t1 &amp; 1\n\\end{array}\\right]\\left[\\begin{array}{c}\n\t2 \\\\ 1\n\\end{array}\\right] = \\left[\\begin{array}{c}\n\t4 \\\\ 3\n\\end{array}\\right]\n$$\nrelative to the basis $B$.  Since\n$$\nP^{-1} = \\left[\\begin{array}{cc}\n\t\\frac{1}{5} &amp; \\frac{2}{5} \\\\\n\t-\\frac{1}{5} &amp; \\frac{3}{5}\n\\end{array}\\right],\n$$\nwe can verify that \n$$\n[{\\bf v}]_{B&#8217;} = \\left[\\begin{array}{cc}\n\t\\frac{1}{5} &amp; \\frac{2}{5} \\\\\n\t-\\frac{1}{5} &amp; \\frac{3}{5}\n\\end{array}\\right]\\left[\\begin{array}{c}\n\t4 \\\\ 3\n\\end{array}\\right] = \\left[\\begin{array}{c}\n\t2 \\\\ 1\n\\end{array}\\right]\n$$\nwhich is what we started with.\n\n<\/p>\n\n\n\n<p>\nIn the following example, we introduce a third basis to look at the\nrelationship between two <i>non-standard<\/i> bases.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/3bases.gif?resize=300%2C300&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-336\"\/><\/figure><\/div>\n\n\n\n<p> Let $B&#8221; = \\left\\{ \\left[ {2 \\atop 1} \\right],\\left[ {1 \\atop 4} \\right]\\right\\}$.  To find the change of coordinates matrix from the basis $B&#8217;$ of the previous example to $B&#8221;$, we first express the basis vectors $\\left[ {3 \\atop 1} \\right]$ and $\\left[ {-2 \\atop 1} \\right]$ of $B&#8217;$ as linear combinations of the basis vectors $\\left[  {2 \\atop 1} \\right]$ and $\\left[ {1 \\atop 4} \\right]$ of $B&#8221;$: \\begin{eqnarray*} \\mbox{Set }\\left[ \\begin{array}{c} 3 \\\\ 1 \\end{array}\\right] &amp; = &amp; a\\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] + b\\left[\\begin{array}{c} 1 \\\\ 4 \\end{array}\\right] \\\\ \\left[\\begin{array}{c} -2 \\\\ 1 \\end{array}\\right] &amp; = &amp; c \\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] + d\\left[\\begin{array}{c} 1 \\\\ 4 \\end{array}\\right] \\end{eqnarray*} and solve the resulting systems of r $a,b,c,$ and $d$: \\begin{eqnarray*} \\left[ \\begin{array}{c} 3 \\\\ 1 \\end{array}\\right] &amp; = &amp; \\frac{11}{7} \\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] &#8211; \\frac{1}{7}\\left[\\begin{array}{c} 1 \\\\ 4 \\end{array}\\right] \\\\ \\left[\\begin{array}{c} -2 \\\\ 1 \\end{array}\\right] &amp; = &amp; \\frac{-9}{7}\\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] + \\frac{4}{7}\\left[\\begin{array}{c} 1 \\\\ 4 \\end{array}\\right]  \\end{eqnarray*} Thus, the transition matrix form $B&#8217;$ to $B&#8221;$ is $$ \\left[\\begin{array}{cc} \\frac{11}{7} &amp; \\frac{-9}{7} \\\\ \\frac{-1}{7} &amp; \\frac{4}{7} \\end{array}\\right]. $$ The vector ${\\bf v}$ with coordinates $\\left[ {2 \\atop 1} \\right]$ relative to the basis $B&#8217;$ has coordinates $$ \\left[\\begin{array}{cc} \\frac{11}{7} &amp; \\frac{-9}{7} \\\\ \\frac{-1}{9} &amp; \\frac{4}{7} \\end{array}\\right]\\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] = \\left[\\begin{array}{c} \\frac{13}{7} \\\\ \\frac{2}{7} \\end{array}\\right] $$ relative to the basis $B&#8221;$.  This is, back in the standard basis, $$ [ {\\bf v} ]_B = \\frac{13}{7}\\left[\\begin{array}{c} 2 \\\\ 1 \\end{array}\\right] + \\frac{2}{7}\\left[\\begin{array}{c} 1 \\\\ 4 \\end{array}\\right] = \\left[\\begin{array}{c} 4 \\\\ 3 \\end{array}\\right], $$ which agrees with the results of the previous example. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Rotation of the Coordinate Axes<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/rotation.gif?resize=300%2C300&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-337\"\/><\/figure><\/div>\n\n\n\n<p>\n\nSuppose we obtain a new coordinate system from the standard\nrectangular coordinate system by rotating the axes counterclockwise by\nan angle $\\theta$.  The new basis $B&#8217; = \\left\\{{\\bf u&#8217;, v&#8217;}\\right\\}$\nof unit vectors along the $x&#8217;$- and $y&#8217;$-axes, respectively, has\ncoordinates\n\\begin{eqnarray*}\n\t~[{\\bf u&#8217;}]_B  &amp; = &amp; \\left[\\begin{array}{c}\n\t\t\\cos\\theta \\\\ \\sin\\theta\n\t\\end{array}\\right] \\\\\n\t~[{\\bf v&#8217;}]_B &amp; = &amp; \\left[\\begin{array}{c}\n\t\t-\\sin\\theta \\\\ \\cos\\theta\n\t\\end{array} \\right]\n\\end{eqnarray*}\nin the original coordinate system.  \n\n<\/p>\n\n\n\n<p>\nThus, $P = \\left[ \\begin{array}{cc}\n\t\\cos\\theta &amp; -\\sin\\theta \\\\\n\t\\sin\\theta &amp; \\cos\\theta\n\\end{array}\\right]$ and $P^{-1} = \\left[ \\begin{array}{cc}\n\t\\cos\\theta &amp; \\sin\\theta \\\\\n\t-\\sin\\theta &amp; \\cos\\theta\n\\end{array}\\right]$.  A vector $\\left[ {x \\atop y} \\right]_B$ in the\noriginal coordinate system has coordinates $\\left[ {x&#8217; \\atop y&#8217;}\n\\right]_{B&#8217;}$ given by\n$$\n\\left[\\begin{array}{c}\n\tx&#8217; \\\\ y&#8217;\n\\end{array}\\right]_{B&#8217;} = \\left[\\begin{array}{cc}\n\t\\cos\\theta &amp; \\sin\\theta \\\\\n\t-\\sin\\theta &amp; \\cos\\theta\n\\end{array}\\right]\\left[\\begin{array}{c}\n\tx \\\\ y\n\\end{array}\\right]_{B}\n$$\nin the rotated coordinate system.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/rotation_example.gif?resize=300%2C300&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-338\"\/><\/figure><\/div>\n\n\n\n<p>\n\nThe vector $[{\\bf v}]_{B}= \\left[ {3 \\atop 2} \\right]$ in the original\ncoordinate system has coordinates\n$$\n[{\\bf v}]_{B&#8217;} = \\left[\\begin{array}{cc}\n\t\\frac{\\sqrt{2}}{2} &amp; \\frac{\\sqrt{2}}{2} \\\\\n\t-\\frac{\\sqrt{2}}{2} &amp; \\frac{\\sqrt{2}}{2}\n\\end{array}\\right]\\left[\\begin{array}{c}\n\t3 \\\\ 2\n\\end{array}\\right] = \\left[\\begin{array}{c}\n\t\\frac{5\\sqrt{2}}{2} \\\\ -\\frac{\\sqrt{2}}{2}\n\\end{array}\\right]\n$$\nin the coordinate system formed by rotating the axes by $45^{\\circ}$.\n\n<\/p>\n\n\n\n<p>\nIn the following Exploration, set up your own basis in $R^2$ and\ncompare the coordinates of vectors in your basis to their coordinates\nin the standard basis.\n\n<\/p>\n\n\n\n<p>Exploration\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center><h4 class=\"wp-block-heading\">Key Concepts<\/h4><\/center>\n\n\n\n<p>\n\nLet $B = \\{{\\bf u,v}\\}$ and $B&#8217; = \\{ {\\bf u&#8217;, v&#8217;} \\}$ be two bases for\n$R^2$.  If $[{\\bf u}]_{B}= \\left[ {a \\atop b} \\right]$ and $[{\\bf\nv}]_{B}= \\left[ {c \\atop d} \\right]$, then $P = \\left[\\begin{array}{cc}\n\ta &amp; c \\\\\n\tb &amp; d\n\\end{array}\\right]$ is the <\/p>\n\n\n\n<b>change of coordinates matrix<\/b>\n\n\n\n<p> from\n$B&#8217;$ to $B$ and $P^{-1}$ is the change of coordinates matrix from $B$\nto $B&#8217;$.  That is, for any ${\\bf v} \\in V$, \n\\begin{eqnarray*}\n\t~[{\\bf v}]_{B} &amp; = &amp; P[{\\bf v}]_{B&#8217;}  \\\\\n\t~[{\\bf v}]_{B&#8217;} &amp; = &amp;  P^{-1}[{\\bf v}]_{B}.\n\\end{eqnarray*}\n\n<br><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ4010\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Change of Basis &#8211; HMC Calculus Tutorial Let $V$ be a vector space and let $S = \\{{\\bf v_1,v_2, \\ldots, v_n}\\}$ be a set of vectors in $V$. Recall that $S$ forms a basis for $V$ if the following two conditions hold: Let $V$ be a vector space and let $S = \\{{\\bf v_1,v_2, \\ldots,&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-221","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/221","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=221"}],"version-history":[{"count":23,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/221\/revisions"}],"predecessor-version":[{"id":1255,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/221\/revisions\/1255"}],"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=221"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}