{"id":225,"date":"2019-09-10T17:31:12","date_gmt":"2019-09-10T17:31:12","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=225"},"modified":"2020-06-18T16:54:08","modified_gmt":"2020-06-18T16:54:08","slug":"geometry-of-linear-transformations","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/linear-algebra\/geometry-of-linear-transformations\/","title":{"rendered":"Geometry of Linear Transformations"},"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>Geometry of Linear Transformations of the Plane &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>Let $V$ and $W$ be <strong>vector spaces<\/strong>.  <\/p>\n\n\n<style>.kt-accordion-id_4a066e-03 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_4a066e-03 .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_4a066e-03 > .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_4a066e-03: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_4a066e-03: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 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.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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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 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.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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03: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_4a066e-03 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_4a066e-03 .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_4a066e-03 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_72cb16-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\">Definition of Vector Spaces<\/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\nA <strong>vector space<\/strong> $V$ consists of a set of elements (called <strong>vectors<\/strong>) on which we have two different operations, usually thought of as vector addition and scalar multiplication,  satisfying a prescribed set of properties. \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><br>Recall that a function $T:V \\rightarrow W$ is called a <strong>linear transformation<\/strong> if it preserves both vector addition and scalar multiplication:  \\begin{eqnarray*} T({\\bf v_1}+ {\\bf v_2}) &amp; = &amp; T({\\bf v_1}) +  T({\\bf v_2}) \\\\ T(r{\\bf v_1}) &amp; = &amp; rT({\\bf v_1})   \\end{eqnarray*} <\/p>\n\n\n\n<p>\n\tfor all ${\\bf v_1, v_2} \\in V$. $\\qquad\\qquad\\qquad\\qquad$\n<\/p>\n\n\n\n<p>\nIf $V = R^{2}$ and $W = R^{2}$, then $T:R^2 \\rightarrow R^2$ is a\nlinear transformation if and only if there exists a $2 \\times 2$\nmatrix $A$ such that $T({\\bf v}) = A{\\bf v}$ for all ${\\bf v} \\in\nR^2$.  Matrix $A$ is called the <b>standard matrix<\/b> for $T$.  The\ncolumns of $A$ are $T \\left( \\left[ {1 \\atop 0} \\right] \\right)$ and \n$T \\left( \\left[ {0 \\atop 1} \\right] \\right)$, respectively.  Since\neach linear transformation of the plane has a unique standard matrix,\nwe will identify linear transformations of the plane by their standard\nmatrices.  It can be shown that if $A$ is invertible, then the linear\ntransformation defined by $A$ maps parollelograms to parallelograms.\nWe will often illustrate the action of a linear transformation $T:R^2\n\\rightarrow R^2$ by looking at the image of a unit square under $T$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Rotations<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"250\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/geom1.gif?resize=325%2C250\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-359\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\tThe standard matrix for the linear transformation $T:R^2 \\rightarrow\n\t\tR^2$ that rotates vectors by an angle $\\theta$ is\n\t\t$$\n\t\tA = \\left[\\begin{array}{cc}\n\t\t\t\\cos\\theta &amp; -\\sin\\theta \\\\\n\t\t\t\\sin\\theta &amp; \\cos\\theta\n\t\t\\end{array} \\right].\n\t\t$$\n\t\tThis is easily drived by noting that\n\t\t\\begin{eqnarray*}\n\t\t\tT\\left( \\left[ {1 \\atop 0} \\right] \\right) &amp; = &amp; \\left[ {\\cos\\theta\n\t\t\t \\atop \\sin\\theta} \\right] \\\\\n\t\t\tT\\left( \\left[ {0 \\atop 1} \\right] \\right) &amp; = &amp; \\left[ {-\\sin\\theta\n\t\t\t \\atop \\cos\\theta} \\right].\n\t\t\\end{eqnarray*}\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Reflections<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"125\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/geom3.gif?resize=325%2C125&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-361\"\/><\/figure><\/div>\n\n\n\n<p>For every line in the plane, there is a linear transformation that reflects vectors about that line.  Reflection about the $x$-axis is given by the standard matrix $$ A = \\left[ \\begin{array}{cc} 1 &amp; 0\\\\ 0 &amp; -1 \\end{array} \\right] $$ which takes the vector $\\left[ {x \\atop y} \\right]$ to $\\left[ {x \\atop -y} \\right]$.  Reflection about the $y$-axis is given by the standard matrix $$ A = \\left[ \\begin{array}{cc} -1 &amp; 0\\\\ 0 &amp; 1 \\end{array} \\right] $$ taking $\\left[ {x \\atop y} \\right]$ to $\\left[ {-x \\atop y} \\right]$. Finally, reflection about the line $y=x$ is given by $$ A = \\left[ \\begin{array}{cc} 0 &amp; 1 \\\\ 1 &amp; 0 \\end{array} \\right] $$ and takes the vector $\\left[ {x \\atop y} \\right]$ to $\\left[ {y \\atop x} \\right]$. <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"125\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/geom2.gif?resize=325%2C125&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-360\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"125\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/geom4.gif?resize=325%2C125&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-362\"\/><\/figure><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Expansions and Compressions<\/h4>\n\n\n\n<p> The standard matrix $$ A = \\left[ \\begin{array}{cc} k &amp; 0 \\\\ 0 &amp; 1 \\end{array} \\right] $$ &#8220;stretches&#8221; the vector $\\left[ {x \\atop y} \\right]$ along the $x$-axis to $\\left[ {kx \\atop y} \\right]$ for $k &gt; 1$ and &#8220;compresses&#8221; it along the $x$-axis for $0~ &lt; ~ k ~ &lt; ~ 1$. <\/p>\n\n\n\n<ul class=\"wp-block-gallery columns-2 is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\"><li class=\"blocks-gallery-item\"><figure><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"250\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/geom6.gif?resize=325%2C250&#038;ssl=1\" alt=\"Compression in the y-axis\" data-id=\"364\" data-link=\"https:\/\/math.hmc.edu\/calculus\/geom6\/\" class=\"wp-image-364\"\/><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"250\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/geom5.gif?resize=325%2C250&#038;ssl=1\" alt=\"Compression in the x-direction\" data-id=\"363\" data-link=\"https:\/\/math.hmc.edu\/calculus\/geom5\/\" class=\"wp-image-363\"\/><\/figure><\/li><\/ul>\n\n\n\n<p>\n\t\tSimilarlarly,\n\t\t$$\n\t\tA = \\left[ \\begin{array}{cc}\n\t\t\t1 &amp; 0 \\\\\n\t\t\t0 &amp; k\n\t\t\\end{array} \\right]\n\t\t$$\n\t\tstretches or compresses vectors $\\left[ {x \\atop y} \\right]$ to\n\t\t$\\left[ {x \\atop ky} \\right]$ along the $y$-axis.\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Shears<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"125\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/geom7.gif?resize=325%2C125\" alt=\"A shear in the x-direction\" class=\"wp-image-365\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\tThe standard matrix\n\t\t$$\n\t\tA = \\left[ \\begin{array}{cc}\n\t\t\t1 &amp; k \\\\\n\t\t\t0 &amp; 1\n\t\t\\end{array} \\right]\n\t\t$$\n\t\ttaking vectors $\\left[ {x \\atop y} \\right]$ to $\\left[ {x+ky \\atop y}\n\t\t\\right]$ is called a <b>shear in the $x$-direction<\/b>.\n<\/p>\n\n\n\n<ul class=\"wp-block-gallery alignright columns-1 is-cropped wp-block-gallery-2 is-layout-flex wp-block-gallery-is-layout-flex\"><li class=\"blocks-gallery-item\"><figure><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"125\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/geom8.gif?resize=325%2C125\" alt=\"A shear in the y-direction\" data-id=\"366\" class=\"wp-image-366\"\/><\/figure><\/li><\/ul>\n\n\n\n<p>\n\t\tSimilarly,\n\t\t$$\n\t\tA = \\left[ \\begin{array}{cc}\n\t\t\t1 &amp; 0 \\\\\n\t\t\tk &amp; 1\n\t\t\\end{array} \\right]\n\t\t$$\n\t\ttakes vectors $\\left[ {x \\atop y} \\right]$ to $\\left[ {x \\atop y+kx}\n\t\t\\right]$ and is called a <b>shear in the $y$-direction<\/b>.\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> If finitely many linear transformations from $R^2$ to $R^2$ are performed in succession, then there exists a single linear transformation with thte same effect. <br><br> <\/li><li> If the standard matrix for a linear transformation $T: R^2 \\rightarrow R^2$ is <b>invertible<\/b>,  such that   <br>$$  AA^{-1} = A^{-1} A = I. $$<br>Then it can be shown that the geometric effect of $T$ is the same as some sequence of reflections, expansions, compressions, and shears.<br>Note:  For a $2 \\times 2$ matrix, $A$ is invertible if and only if $\\det A \\neq 0$.   <\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<p> For every linear transformation $T: R^2 \\rightarrow R^2$ of the plane, there exists a standard matrix $A$ such that  $$ T({\\bf v}) = A{\\bf v} {\\small\\textrm{ for all }} {\\bf v} \\in R^2. $$ Every linear transformation of the plane with an <em>invertible  <\/em>standard matrix has the geometric effect of a sequence of reflections, expansions, compressions, and shears.  <\/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\/QZ4110\/\">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>Geometry of Linear Transformations of the Plane &#8211; HMC Calculus Tutorial Let $V$ and $W$ be vector spaces. Recall that a function $T:V \\rightarrow W$ is called a linear transformation if it preserves both vector addition and scalar multiplication: \\begin{eqnarray*} T({\\bf v_1}+ {\\bf v_2}) &amp; = &amp; T({\\bf v_1}) + T({\\bf v_2}) \\\\ T(r{\\bf v_1})&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-225","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/225","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=225"}],"version-history":[{"count":10,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/225\/revisions"}],"predecessor-version":[{"id":1235,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/225\/revisions\/1235"}],"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=225"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}