-
Notifications
You must be signed in to change notification settings - Fork 0
/
TransformationExamples.html
144 lines (141 loc) · 14.1 KB
/
TransformationExamples.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
<!DOCTYPE html>
<!--********************************************-->
<!--* Generated from PreTeXt source *-->
<!--* on 2021-08-31T10:06:24-05:00 *-->
<!--* A recent stable commit (2020-08-09): *-->
<!--* 98f21740783f166a773df4dc83cab5293ab63a4a *-->
<!--* *-->
<!--* https://pretextbook.org *-->
<!--* *-->
<!--********************************************-->
<html lang="en-US">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Rotations, reflections, projections and dilations</title>
<meta name="Keywords" content="Authored in PreTeXt">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script src="https://sagecell.sagemath.org/embedded_sagecell.js"></script><script>window.MathJax = {
tex: {
inlineMath: [['\\(','\\)']],
tags: "none",
useLabelIds: true,
tagSide: "right",
tagIndent: ".8em",
packages: {'[+]': ['base', 'extpfeil', 'ams', 'amscd', 'newcommand', 'knowl']}
},
options: {
ignoreHtmlClass: "tex2jax_ignore",
processHtmlClass: "has_am",
renderActions: {
findScript: [10, function (doc) {
document.querySelectorAll('script[type^="math/tex"]').forEach(function(node) {
var display = !!node.type.match(/; *mode=display/);
var math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
var text = document.createTextNode('');
node.parentNode.replaceChild(text, node);
math.start = {node: text, delim: '', n: 0};
math.end = {node: text, delim: '', n: 0};
doc.math.push(math);
});
}, '']
},
},
chtml: {
scale: 0.88,
mtextInheritFont: true
},
loader: {
load: ['input/asciimath', '[tex]/extpfeil', '[tex]/amscd', '[tex]/newcommand', '[pretext]/mathjaxknowl3.js'],
paths: {pretext: "https://pretextbook.org/js/lib"},
},
};
</script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.sticky.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.espy.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext_add_on.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script xmlns:svg="http://www.w3.org/2000/svg">sagecellEvalName='Evaluate (Sage)';
</script><link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext_add_on.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/banner_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/toc_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/knowls_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/style_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/colors_brown_gold.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/setcolors.css" rel="stylesheet" type="text/css">
<!-- 2019-10-12: Temporary - CSS file for experiments with styling --><link xmlns:svg="http://www.w3.org/2000/svg" href="developer.css" rel="stylesheet" type="text/css">
</head>
<body class="pretext-book has-toc has-sidebar-left">
<a class="assistive" href="#content">Skip to main content</a><div xmlns:svg="http://www.w3.org/2000/svg" id="latex-macros" class="hidden-content" style="display:none">\(\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Q{{\mathbb Q}}
\def\Z{{\mathbb Z}}
\def\N{{\mathbb N}}
\def\vec#1{\mathbf #1}
\newcommand{\adj}{\mathop{\mathrm{adj}}}
\newcommand{\diag}{\mathop{\mathrm{diag}}}
\newcommand{\proj}{\mathop{\mathrm{proj}}}
\newcommand{\Span}{\mathop{\mathrm{span}}}
\newcommand{\sgn}{\mathop{\mathrm{sgn}}}
\newcommand{\tr}{\mathop{\mathrm{tr}}}
\newcommand{\rowint}[2]{R_{#1} \leftrightarrow R_{#2}}
\newcommand{\rowmul}[2]{R_{#1}\gets {#2}R_{#1}}
\newcommand{\rowadd}[3]{R_{#1}\gets R_{#1}+#2R_{#3}}
\newcommand{\rowsub}[3]{R_{#1}\gets R_{#1}-#2R_{#3}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\)</div>
<header id="masthead" class="smallbuttons"><div class="banner"><div class="container">
<a id="logo-link" href="http://www.umanitoba.ca" target="_blank"><img src="images/umlogo.png" alt="Logo image"></a><div class="title-container">
<h1 class="heading"><a href="mblinalg.html"><span class="title">Manitoba linear algebra</span></a></h1>
<p class="byline">Michael Doob</p>
</div>
</div></div>
<nav xmlns:svg="http://www.w3.org/2000/svg" id="primary-navbar" class="navbar"><div class="container">
<div class="navbar-top-buttons">
<button class="sidebar-left-toggle-button button active" aria-label="Show or hide table of contents sidebar">Contents</button><div class="tree-nav toolbar toolbar-divisor-3"><span class="threebuttons"><a id="previousbutton" class="previous-button toolbar-item button" href="section-40.html" title="Previous">Prev</a><a id="upbutton" class="up-button button toolbar-item" href="LinearTransformations.html" title="Up">Up</a><a id="nextbutton" class="next-button button toolbar-item" href="LinearTransformationIntroduction.html" title="Next">Next</a></span></div>
</div>
<div class="navbar-bottom-buttons toolbar toolbar-divisor-4">
<button class="sidebar-left-toggle-button button toolbar-item active">Contents</button><a class="previous-button toolbar-item button" href="section-40.html" title="Previous">Prev</a><a class="up-button button toolbar-item" href="LinearTransformations.html" title="Up">Up</a><a class="next-button button toolbar-item" href="LinearTransformationIntroduction.html" title="Next">Next</a>
</div>
</div></nav></header><div class="page">
<div xmlns:svg="http://www.w3.org/2000/svg" id="sidebar-left" class="sidebar" role="navigation"><div class="sidebar-content">
<nav id="toc"><ul>
<li class="link frontmatter"><a href="Frontmatter.html" data-scroll="Frontmatter"><span class="title">Title Page</span></a></li>
<li class="link"><a href="SysLinEq.html" data-scroll="SysLinEq"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations</span></a></li>
<li class="link"><a href="MatrixTheoryIntro.html" data-scroll="MatrixTheoryIntro"><span class="codenumber">2</span> <span class="title">Matrix Theory</span></a></li>
<li class="link"><a href="Determinants.html" data-scroll="Determinants"><span class="codenumber">3</span> <span class="title">The Determinant</span></a></li>
<li class="link"><a href="EuclideanSpace.html" data-scroll="EuclideanSpace"><span class="codenumber">4</span> <span class="title">Vectors in Euclidean \(n\) space</span></a></li>
<li class="link"><a href="chapter-5.html" data-scroll="chapter-5"><span class="codenumber">5</span> <span class="title">Eigenvalues and eigenvectors</span></a></li>
<li class="link"><a href="LinearTransformations.html" data-scroll="LinearTransformations"><span class="codenumber">6</span> <span class="title">Linear transformations</span></a></li>
<li class="link"><a href="ExtraTopics.html" data-scroll="ExtraTopics"><span class="codenumber">7</span> <span class="title">Additional Topics</span></a></li>
</ul></nav><div class="extras"><nav><a class="pretext-link" href="https://pretextbook.org">Authored in PreTeXt</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://www.mathjax.org/badge/badge.gif" alt="Powered by MathJax"></a></nav></div>
</div></div>
<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="TransformationExamples"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">6.2</span> <span class="title">Rotations, reflections, projections and dilations</span>
</h2>
<section class="subsection" id="subsection-90"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">6.2.1</span> <span class="title">Transformations \(T\colon \R^2\to\R^2\)</span>
</h3>
<p id="p-1195">We wish to consider some transformations that are geometrically inspired. The following examples from \(\R^2\) will be useful as we study linear transformations.</p>
<article class="example example-like" id="PlaneRotation"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-PlaneRotation"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">6.2.1</span><span class="period">.</span><span class="space"> </span><span class="title">A rotation in the plane.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-PlaneRotation"><article class="example example-like"><p id="p-1196">We start with a point \(\vec x\) in the plane and rotate it through an angle \(\theta\) counterclockwise around the origin. This new point is \(L(\vec x)\text{.}\)</p>
<figure class="figure figure-like" id="figure-60"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;"><div class="asymptote-box" style="padding-top: 112.16707110817%"><iframe src="images/image-62.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">6.2.2<span class="period">.</span></span><span class="space"> </span>The vector \(\vec x\) rotated to \(L(\vec x)\) though an angle \(\theta\)</figcaption></figure></article></div>
<article class="example example-like" id="PlaneReflection"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-PlaneReflection"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">6.2.3</span><span class="period">.</span><span class="space"> </span><span class="title">A reflection in the plane.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-PlaneReflection"><article class="example example-like"><p id="p-1197">We start with a point \(\vec x\) in the plane and reflect in across the line with equation \(y=x\text{.}\) This new point is \(L(\vec x)\text{.}\)</p>
<figure class="figure figure-like" id="figure-61"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;"><div class="asymptote-box" style="padding-top: 101.773442399819%"><iframe src="images/image-63.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">6.2.4<span class="period">.</span></span><span class="space"> </span>The reflection of the vector \(\vec x\) by the line \(y=x\)</figcaption></figure></article></div>
<article class="example example-like" id="PlaneProjection"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-PlaneProjection"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">6.2.5</span><span class="period">.</span><span class="space"> </span><span class="title">A projection in the plane.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-PlaneProjection"><article class="example example-like"><p id="p-1198">We start with a point \(\vec x\) in the plane and drop a perpendicular to the line with equation \(y=x\text{.}\) This new point is \(L(\vec x)\text{.}\)</p>
<figure class="figure figure-like" id="figure-62"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;"><div class="asymptote-box" style="padding-top: 101.773442399819%"><iframe src="images/image-64.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">6.2.6<span class="period">.</span></span><span class="space"> </span></figcaption></figure></article></div>
<article class="example example-like" id="PlaneDilation"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-PlaneDilation"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">6.2.7</span><span class="period">.</span><span class="space"> </span><span class="title">A dilation in the plane.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-PlaneDilation"><article class="example example-like"><p id="p-1199">We start with a real number \(s\gt 0\text{.}\) For a point \(\vec x\text{,}\) let \(L(\vec x)=s\vec x\text{.}\) If \(\vec x\not=\vec0\text{,}\) then \(L(\vec x)\) is on the line joining \(\vec x\) and \(\vec 0\text{.}\) The distance from \(\vec x\) to \(\vec 0\) has been stretched by a factor of \(s\) to get the distance from \(L(\vec x)\) to \(\vec 0\text{.}\)</p>
<figure class="figure figure-like" id="figure-63"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;"><div class="asymptote-box" style="padding-top: 99.5882480957563%"><iframe src="images/image-65.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">6.2.8<span class="period">.</span></span><span class="space"> </span></figcaption></figure></article></div></section></section></div></main>
</div>
</body>
</html>