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<div id="jb-print-docs-body" class="onlyprint">
    <h1>The Matrix of a Linear Transformation</h1>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#geometric-linear-transformations-of-mathbb-r-2">Geometric Linear Transformations of <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span></a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#area-is-scaled-by-the-determinant">Area is Scaled by the Determinant</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#existence-and-uniqueness">Existence and Uniqueness</a></li>
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  <section class="tex2jax_ignore mathjax_ignore" id="the-matrix-of-a-linear-transformation">
<h1>The Matrix of a Linear Transformation<a class="headerlink" href="#the-matrix-of-a-linear-transformation" title="Permalink to this heading">#</a></h1>
<p>In the last lecture we introduced the idea of a <strong>linear transformation</strong>:</p>
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<p>We have seen that every matrix multiplication is a linear transformation from vectors to vectors.</p>
<p>But, are there any other possible linear transformations from vectors to vectors?</p>
<p>No.</p>
<p>In other words, the reverse statement is also true:</p>
<blockquote>
<div><p><font color = "blue"> every linear transformation from vectors to vectors is a matrix multiplication. </font></p>
</div></blockquote>
<p>We’ll now prove this fact.</p>
<p>We’ll do it <strong>constructively</strong>, meaning we’ll actually show how to find the matrix corresponding to any given linear transformation <span class="math notranslate nohighlight">\(T\)</span>.</p>
<p><strong>Theorem.</strong>  Let <span class="math notranslate nohighlight">\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\)</span> be a linear transformation.   Then there is (always) a unique matrix <span class="math notranslate nohighlight">\(A\)</span> such that:</p>
<div class="math notranslate nohighlight">
\[ T({\bf x}) = A{\bf x} \;\;\; \mbox{for all}\; {\bf x} \in \mathbb{R}^n.\]</div>
<p>In fact, <span class="math notranslate nohighlight">\(A\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix whose <span class="math notranslate nohighlight">\(j\)</span>th column is the vector <span class="math notranslate nohighlight">\(T({\bf e_j})\)</span>, where <span class="math notranslate nohighlight">\({\bf e_j}\)</span> is the <span class="math notranslate nohighlight">\(j\)</span>th column of the identity matrix in <span class="math notranslate nohighlight">\(\mathbb{R}^n\)</span>:</p>
<div class="math notranslate nohighlight">
\[A = \left[T({\bf e_1}) \dots T({\bf e_n})\right].\]</div>
<p><span class="math notranslate nohighlight">\(A\)</span> is called the <em>standard matrix</em> of <span class="math notranslate nohighlight">\(T\)</span>.</p>
<p><strong>Proof.</strong>  Write</p>
<div class="math notranslate nohighlight">
\[{\bf x} = I{\bf x} = \left[{\bf e_1} \dots {\bf e_n}\right]\bf x\]</div>
<div class="math notranslate nohighlight">
\[ = x_1{\bf e_1} + \dots + x_n{\bf e_n}.\]</div>
<p>In other words, for any <span class="math notranslate nohighlight">\(\mathbf{x}\)</span>, we can always expand it as:</p>
<div class="math notranslate nohighlight">
\[\begin{split} \mathbf{x} 
\;= \;\;\;\;\;
\begin{bmatrix}
1 &amp; 0 &amp; \dots &amp; 0 \\
0 &amp; 1 &amp; \dots &amp; 0\\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
0 &amp; 0 &amp; \dots &amp; 1 
\end{bmatrix} \; \begin{bmatrix}x_1\\x_2\\ \vdots \\ x_n\end{bmatrix} 
\;\;\;\;\;= \;\;\;\;\;
\begin{bmatrix} x_1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} +
\begin{bmatrix} 0 \\ x_2 \\ \vdots \\ 0 \end{bmatrix} +
\dots +
\begin{bmatrix} 0 \\ 0 \\ \vdots \\ x_n \end{bmatrix} \end{split}\]</div>
<p>Because <span class="math notranslate nohighlight">\(T\)</span> is linear, we have:</p>
<div class="math notranslate nohighlight">
\[ T({\bf x}) = T(x_1{\bf e_1} + \dots + x_n{\bf e_n})\]</div>
<div class="math notranslate nohighlight">
\[ = x_1T({\bf e_1}) + \dots + x_nT({\bf e_n})\]</div>
<div class="math notranslate nohighlight">
\[\begin{split} = \left[T({\bf e_1}) \dots T({\bf e_n})\right] \, \left[\begin{array}{r}x_1\\\vdots\\x_n\end{array}\right] = A{\bf x}.\end{split}\]</div>
<p>So … we see that the ideas of <strong>matrix multiplication</strong> and <strong>linear transformation</strong> are essentially equivalent when applied to vectors.</p>
<p>Every matrix multiplication is a linear transformation, and every linear transformation from vectors to vectors is a matrix multiplication.</p>
<p><strong>However</strong>, term <em>linear transformation</em> focuses on a <strong>property</strong> of the mapping, while the term <em>matrix multiplication</em> focuses on how such a mapping is <strong>implemented.</strong></p>
<p>This proof shows us an important idea:</p>
<p><font color='blue'> To find the standard matrix of a linear transformation, ask what the transformation does to the columns of <span class="math notranslate nohighlight">\(I\)</span>.</font></p>
<p>In other words, if <span class="math notranslate nohighlight">\( T(\mathbf{x}) = A\mathbf{x} \)</span>, then:</p>
<div class="math notranslate nohighlight">
\[A = \left[T({\bf e_1}) \dots T({\bf e_n})\right].\]</div>
<p>This gives us a way to compute the standard matrix of a transformation.</p>
<p>Now, in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>, <span class="math notranslate nohighlight">\(I = \left[\begin{array}{cc}1&amp;0\\0&amp;1\end{array}\right]\)</span>.   So:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\mathbf{e_1} = \left[\begin{array}{c}1\\0\end{array}\right]\;\;\mbox{and}\;\;\mathbf{e_2} = \left[\begin{array}{c}0\\1\end{array}\right].\end{split}\]</div>
<p>So to find the matrix of any given linear transformation of vectors in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>, we only have to know what that transformation does to these two points:</p>
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<p>This is a <strong>hugely</strong> powerful tool.</p>
<p>Let’s say we start from some given linear transformation; we can use this idea to find the matrix that implements that linear transformation.</p>
<p>For example, let’s consider rotation about the origin as a kind of transformation.</p>
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</div>
</div>
<p>First things first: Is rotation a <strong>linear</strong> transformation?</p>
<p>Recall that a for a transformation to be linear, it must be true that <span class="math notranslate nohighlight">\(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}).\)</span></p>
<p>I’m going to show you a “geometric proof.”</p>
<p>This figure shows that “the rotation of <span class="math notranslate nohighlight">\(\mathbf{u+v}\)</span> is the sum of the rotation of <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> and the rotation of <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>”.</p>
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<p>OK, so rotation is a linear transformation.</p>
<p>Let’s see how to <strong>compute</strong> the linear transformation that is a rotation.</p>
<p><strong>Specifically:</strong>  Let <span class="math notranslate nohighlight">\(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\)</span> be the transformation that rotates each point in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span> about the origin through an angle <span class="math notranslate nohighlight">\(\theta\)</span>, with counterclockwise rotation for a positive angle.</p>
<p>Let’s find the standard matrix <span class="math notranslate nohighlight">\(A\)</span> of this transformation.</p>
<p><strong>Solution.</strong>  The columns of <span class="math notranslate nohighlight">\(I\)</span> are <span class="math notranslate nohighlight">\({\bf e_1} = \left[\begin{array}{r}1\\0\end{array}\right]\)</span> and <span class="math notranslate nohighlight">\({\bf e_2} = \left[\begin{array}{r}0\\1\end{array}\right].\)</span></p>
<p>Referring to the diagram below, we can see that <span class="math notranslate nohighlight">\(\left[\begin{array}{r}1\\0\end{array}\right]\)</span> rotates into <span class="math notranslate nohighlight">\(\left[\begin{array}{r}\cos\theta\\\sin\theta\end{array}\right],\)</span> and <span class="math notranslate nohighlight">\(\left[\begin{array}{r}0\\1\end{array}\right]\)</span> rotates into <span class="math notranslate nohighlight">\(\left[\begin{array}{r}-\sin\theta\\\cos\theta\end{array}\right].\)</span></p>
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<p>So by the Theorem above,</p>
<div class="math notranslate nohighlight">
\[\begin{split} A = \left[\begin{array}{rr}\cos\theta&amp;-\sin\theta\\\sin\theta&amp;\cos\theta\end{array}\right].\end{split}\]</div>
<p>To demonstrate the use of a rotation matrix, let’s rotate the following shape:</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">note</span> <span class="o">=</span> <span class="n">dm</span><span class="o">.</span><span class="n">mnote</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">note</span><span class="p">)</span>
</pre></div>
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<p>The variable <code class="docutils literal notranslate"><span class="pre">note</span></code> is a array of 26 vectors in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span> that define its shape.</p>
<p>In other words, it is a 2 <span class="math notranslate nohighlight">\(\times\)</span> 26 matrix.</p>
<p>To rotate <code class="docutils literal notranslate"><span class="pre">note</span></code> we need to multiply each column of <code class="docutils literal notranslate"><span class="pre">note</span></code> by the rotation matrix <span class="math notranslate nohighlight">\(A\)</span>.</p>
<p>In Python this can be performed using the <code class="docutils literal notranslate"><span class="pre">&#64;</span></code> operator.</p>
<p>That is, if <code class="docutils literal notranslate"><span class="pre">A</span></code> and <code class="docutils literal notranslate"><span class="pre">B</span></code> are matrices,</p>
<p><code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&#64;</span> <span class="pre">B</span></code></p>
<p>will multiply <code class="docutils literal notranslate"><span class="pre">A</span></code> by every column of <code class="docutils literal notranslate"><span class="pre">B,</span></code> and the resulting vectors will be formed
into a matrix.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">angle</span> <span class="o">=</span> <span class="mi">90</span>
<span class="n">theta</span> <span class="o">=</span> <span class="p">(</span><span class="n">angle</span><span class="o">/</span><span class="mi">180</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
    <span class="p">[[</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)],</span>
     <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)]])</span>
<span class="n">rnote</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">note</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">rnote</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<section id="geometric-linear-transformations-of-mathbb-r-2">
<h2>Geometric Linear Transformations of <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span><a class="headerlink" href="#geometric-linear-transformations-of-mathbb-r-2" title="Permalink to this heading">#</a></h2>
<p>Let’s use our understanding of how to construct linear transformations to look at some specific linear transformations of <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span> to <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>.</p>
<p>First, let’s recall the linear transformation</p>
<div class="math notranslate nohighlight">
\[T(\mathbf{x}) = r\mathbf{x}.\]</div>
<p>With <span class="math notranslate nohighlight">\(r &gt; 1\)</span>, this is a dilation.  It moves every vector further from the origin.</p>
<p>Let’s say the dilation is by a factor of 2.5.</p>
<p>To construct the matrix <span class="math notranslate nohighlight">\(A\)</span> that implements this transformation, we ask: where do <span class="math notranslate nohighlight">\({\bf e_1}\)</span> and <span class="math notranslate nohighlight">\({\bf e_2}\)</span> go?</p>
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<p>Under the action of <span class="math notranslate nohighlight">\(A\)</span>, <span class="math notranslate nohighlight">\(\mathbf{e_1}\)</span> goes to <span class="math notranslate nohighlight">\(\left[\begin{array}{c}2.5\\0\end{array}\right]\)</span> and <span class="math notranslate nohighlight">\(\mathbf{e_2}\)</span> goes to <span class="math notranslate nohighlight">\(\left[\begin{array}{c}0\\2.5\end{array}\right]\)</span>.</p>
<p>So the matrix <span class="math notranslate nohighlight">\(A\)</span> must be <span class="math notranslate nohighlight">\(\left[\begin{array}{cc}2.5&amp;0\\0&amp;2.5\end{array}\right]\)</span>.</p>
<p>Let’s test this out:</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">square</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
    <span class="p">[[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
     <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
    <span class="p">[[</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
     <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">]])</span>
<span class="n">display</span><span class="p">(</span><span class="n">Latex</span><span class="p">(</span><span class="sa">rf</span><span class="s2">&quot;$A = </span><span class="si">{</span><span class="n">ltx_array_fmt</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="s1">&#39;</span><span class="si">{:1.1f}</span><span class="s1">&#39;</span><span class="p">)</span><span class="si">}</span><span class="s2">$&quot;</span><span class="p">))</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">square</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">square</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
</pre></div>
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</div>
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<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  2.5 &amp;  0.0\\
  0.0 &amp;  2.5
\end{bmatrix}\end{split}\]</div>
<img alt="_images/424c86d187e8d41231ebf823eb485ccbcf9aa93e11aa72d6e8987e380bb0bb38.png" src="_images/424c86d187e8d41231ebf823eb485ccbcf9aa93e11aa72d6e8987e380bb0bb38.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">note</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">note</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<img alt="_images/25e01f127e433fe6b9d4f903953457a56a4ea5eebb5046c57d46aec29c0cd1ec.png" src="_images/25e01f127e433fe6b9d4f903953457a56a4ea5eebb5046c57d46aec29c0cd1ec.png" />
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<p>OK, now let’s reflect through the <span class="math notranslate nohighlight">\(x_1\)</span> axis.   Where do <span class="math notranslate nohighlight">\({\bf e_1}\)</span> and <span class="math notranslate nohighlight">\({\bf e_2}\)</span> go?</p>
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<img alt="_images/3fd49a4c73bda8df6ccde3282e58e6914f700b9aca1d8b4bd43ba29adfeff9f3.png" src="_images/3fd49a4c73bda8df6ccde3282e58e6914f700b9aca1d8b4bd43ba29adfeff9f3.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span>  <span class="mi">0</span><span class="p">],</span>
     <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
<span class="n">display</span><span class="p">(</span><span class="n">Latex</span><span class="p">(</span><span class="sa">rf</span><span class="s2">&quot;$A = </span><span class="si">{</span><span class="n">ltx_array_fmt</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="s1">&#39;</span><span class="si">{:d}</span><span class="s1">&#39;</span><span class="p">)</span><span class="si">}</span><span class="s2">$&quot;</span><span class="p">))</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">square</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">square</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Reflection through the $x_1$ axis&#39;</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="mi">20</span><span class="p">);</span>
</pre></div>
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</div>
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<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  1  &amp;  0 \\
  0  &amp; -1 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/a16495e2b943b2cf22d5ee385fa85ad236436a4ce483a9f737a77ee5126d2147.png" src="_images/a16495e2b943b2cf22d5ee385fa85ad236436a4ce483a9f737a77ee5126d2147.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">note</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">note</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
</pre></div>
</div>
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<img alt="_images/0ec6657d266118a881cb50025d546b815b892c5d4dd413730200522683251eac.png" src="_images/0ec6657d266118a881cb50025d546b815b892c5d4dd413730200522683251eac.png" />
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<p>What about reflection through the <span class="math notranslate nohighlight">\(x_2\)</span> axis?</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
     <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="n">display</span><span class="p">(</span><span class="n">Latex</span><span class="p">(</span><span class="sa">rf</span><span class="s2">&quot;$A = </span><span class="si">{</span><span class="n">ltx_array_fmt</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="s1">&#39;</span><span class="si">{:2d}</span><span class="s1">&#39;</span><span class="p">)</span><span class="si">}</span><span class="s2">$&quot;</span><span class="p">))</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">square</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotSquare</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">square</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Reflection through the $x_2$ axis&#39;</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="mi">20</span><span class="p">);</span>
</pre></div>
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<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
 -1  &amp;   0 \\
   0  &amp;   1 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/b7cb88d3ffd620a35dca5f7e87d3c3b56064c0b1e289729a20368752a86d647f.png" src="_images/b7cb88d3ffd620a35dca5f7e87d3c3b56064c0b1e289729a20368752a86d647f.png" />
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<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dm</span><span class="o">.</span><span class="n">plotSetup</span><span class="p">()</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">note</span><span class="p">)</span>
<span class="n">dm</span><span class="o">.</span><span class="n">plotShape</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">note</span><span class="p">,</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<img alt="_images/ea3083e8a9b5ec05b2d8eeff534c651826bf73aa0e488d61741d3fb0951bf9fd.png" src="_images/ea3083e8a9b5ec05b2d8eeff534c651826bf73aa0e488d61741d3fb0951bf9fd.png" />
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<p>What about reflection through the line <span class="math notranslate nohighlight">\(x_1 = x_2\)</span>?</p>
<div class="cell tag_remove-input docutils container">
<div class="cell_output docutils container">
<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  0  &amp;  1 \\
  1  &amp;  0 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/529f7f28931bfa043a21c7871bb1d4e1ba3e3dbbb299f175d4a0b25596ef6583.png" src="_images/529f7f28931bfa043a21c7871bb1d4e1ba3e3dbbb299f175d4a0b25596ef6583.png" />
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<img alt="_images/db981e21289a28ddd6ca00a0033342e59730d28804580c1431d571bfaed7708c.png" src="_images/db981e21289a28ddd6ca00a0033342e59730d28804580c1431d571bfaed7708c.png" />
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<p>What about reflection through the line <span class="math notranslate nohighlight">\(x_1 = -x_2\)</span>?</p>
<div class="cell tag_remove-input docutils container">
<div class="cell_output docutils container">
<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  0  &amp; -1 \\
 -1  &amp;  0 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/07b08ed0ecc3af8b530ef383bd96bd151bfe283df0bb2dbe94f480ef7c5d2e30.png" src="_images/07b08ed0ecc3af8b530ef383bd96bd151bfe283df0bb2dbe94f480ef7c5d2e30.png" />
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<p>What about reflection through the origin?</p>
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\[\begin{split}A = \begin{bmatrix}
 -1  &amp;  0 \\
  0  &amp; -1 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/87d9ce11b89482fbd6fbbec3145cc6e4d4ea62a29e6c4a5cdf1607300dd37e22.png" src="_images/87d9ce11b89482fbd6fbbec3145cc6e4d4ea62a29e6c4a5cdf1607300dd37e22.png" />
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\[\begin{split}A = \begin{bmatrix}
  0.45 &amp;  0.00\\
  0.00 &amp;  1.00
\end{bmatrix}\end{split}\]</div>
<img alt="_images/1e843e1262ae7f59960f257cbb4ca4988791d2044d959b07edd7841a07eacc7d.png" src="_images/1e843e1262ae7f59960f257cbb4ca4988791d2044d959b07edd7841a07eacc7d.png" />
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\[\begin{split}A = \begin{bmatrix}
  2.5 &amp;  0.0\\
  0.0 &amp;  1.0
\end{bmatrix}\end{split}\]</div>
<img alt="_images/26433b1ee0b58b772386457e13cf6d072c4026e6de29ab76cd3b19637df00fa2.png" src="_images/26433b1ee0b58b772386457e13cf6d072c4026e6de29ab76cd3b19637df00fa2.png" />
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\[\begin{split}A = \begin{bmatrix}
  1.0 &amp;  0.0\\
 -1.5 &amp;  1.0
\end{bmatrix}\end{split}\]</div>
<img alt="_images/ac531ab366540b8690f9d499c612f0ddbc962ae7121140d3d2befc39d8899f5d.png" src="_images/ac531ab366540b8690f9d499c612f0ddbc962ae7121140d3d2befc39d8899f5d.png" />
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<p>Now let’s look at a particular kind of transformation called a <strong>projection</strong>.</p>
<p>Imagine we took any given point and ‘dropped’ it onto the <span class="math notranslate nohighlight">\(x_1\)</span>-axis.</p>
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\[\begin{split}A = \begin{bmatrix}
  1  &amp;  0 \\
  0  &amp;  0 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/4ba9d4fc56a086125631446d7920f61efea13288c514ad88a49cef89d75f3f61.png" src="_images/4ba9d4fc56a086125631446d7920f61efea13288c514ad88a49cef89d75f3f61.png" />
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<p>What happens to the <strong>shape</strong> of the point set?</p>
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\[\begin{split}A = \begin{bmatrix}
  0  &amp;  0 \\
  0  &amp;  1 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/f1eabbe8b54a65981a170bcfee60a8e88c6182a8cfaf4fefa3dfe62ff98037c7.png" src="_images/f1eabbe8b54a65981a170bcfee60a8e88c6182a8cfaf4fefa3dfe62ff98037c7.png" />
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</section>
<section id="area-is-scaled-by-the-determinant">
<h2>Area is Scaled by the Determinant<a class="headerlink" href="#area-is-scaled-by-the-determinant" title="Permalink to this heading">#</a></h2>
<p>Notice that in some of the transformations above, the “size” of a shape grows or shrinks.</p>
<p>Let’s look at how area (or volume) of a shape is affected by a linear transformation.</p>
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<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  0.45 &amp;  0.00\\
  0.00 &amp;  1.00
\end{bmatrix}\end{split}\]</div>
<img alt="_images/8f7cd1ba90961d6ec300516c1d8cd93fd34ef28437ca1afc81f231dda6bd597d.png" src="_images/8f7cd1ba90961d6ec300516c1d8cd93fd34ef28437ca1afc81f231dda6bd597d.png" />
</div>
</div>
<p>In this transformation, each unit of area in the blue shape is transformed to a smaller region in the red shape.</p>
<p>So to understand how area changes, it suffices to ask what happens to the unit square (or hypercube):</p>
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</div>
<p>Let’s denote the matrix of our linear transformation as:</p>
<div class="math notranslate nohighlight">
\[\begin{split} A = \begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix} \end{split}\]</div>
<p>Then, here is what happens to the unit square:</p>
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<p>Now, let’s determine the area of the blue diamond in terms of <span class="math notranslate nohighlight">\(a, b, c\)</span>, and <span class="math notranslate nohighlight">\(d\)</span>.</p>
<p>To do that, we’ll use this diagram:</p>
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<p>Each of the triangles and rectangles has an area we can determine in terms of <span class="math notranslate nohighlight">\(a, b, c\)</span> and <span class="math notranslate nohighlight">\(d\)</span>.</p>
<p>The large rectangle has sides <span class="math notranslate nohighlight">\((a+b)\)</span> and <span class="math notranslate nohighlight">\((c+d)\)</span>, so its area is:</p>
<div class="math notranslate nohighlight">
\[ (a+b)(c+d) = ac + ad + bc + bd. \]</div>
<p>From this large rectangle we need to subtract:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(bd\)</span> (red triangles),</p></li>
<li><p><span class="math notranslate nohighlight">\(ac\)</span> (gray triangles), and</p></li>
<li><p><span class="math notranslate nohighlight">\(2bc\)</span> (green rectangles).</p></li>
</ul>
<p>So the area of the blue diamond is:</p>
<div class="math notranslate nohighlight">
\[ (ac + ad + bc + bd) - (bd + ac + 2bc) \]</div>
<div class="math notranslate nohighlight">
\[ = ad - bc \]</div>
<p>So we conclude that when we use a linear transformation</p>
<div class="math notranslate nohighlight">
\[\begin{split} A = \begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix} \end{split}\]</div>
<p>the <strong>area of a unit square</strong> (or any shape) <strong>is scaled by a factor of <span class="math notranslate nohighlight">\(ad - bc\)</span>.</strong></p>
<p>This quantity is a fundamental property of the matrix <span class="math notranslate nohighlight">\(A\)</span>.</p>
<p>So, we give it a name: it is the <strong>determinant</strong> of <span class="math notranslate nohighlight">\(A\)</span>.</p>
<p>We denote it as</p>
<div class="math notranslate nohighlight">
\[\det(A)\]</div>
<p>So, for a <span class="math notranslate nohighlight">\(2\times 2\)</span> matrix <span class="math notranslate nohighlight">\( A = \begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix} \)</span>,</p>
<div class="math notranslate nohighlight">
\[\det(A) = ad-bc.\]</div>
<p>However, the determinant can be defined for <strong>any</strong> <span class="math notranslate nohighlight">\(n\times n\)</span> (square) matrix.</p>
<p>For a square matrix <span class="math notranslate nohighlight">\(A\)</span> larger than <span class="math notranslate nohighlight">\(2\times 2\)</span>, the determinant tells us how the <strong>volume</strong> of a unit (hyper)cube is scaled when it is linearly transformed by <span class="math notranslate nohighlight">\(A\)</span>.</p>
<p>We will learn how to compute determinants for larger matrices in a later lecture.</p>
<p>There are important cases in which the determinant of a matrix is <strong>zero.</strong></p>
<p>When does it happen that <span class="math notranslate nohighlight">\(\det(A) = 0\)</span>?</p>
<p>Consider when <span class="math notranslate nohighlight">\(A\)</span> is the matrix of a projection:</p>
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<div class="output text_latex math notranslate nohighlight">
\[\begin{split}A = \begin{bmatrix}
  1  &amp;  0 \\
  0  &amp;  0 
\end{bmatrix}\end{split}\]</div>
<img alt="_images/4ba9d4fc56a086125631446d7920f61efea13288c514ad88a49cef89d75f3f61.png" src="_images/4ba9d4fc56a086125631446d7920f61efea13288c514ad88a49cef89d75f3f61.png" />
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</div>
<p>The unit square has been collapsed onto the <span class="math notranslate nohighlight">\(x\)</span>-axis, resulting in a shape with area of zero.</p>
<p>This is confirmed by the determinant, which is</p>
<div class="math notranslate nohighlight">
\[\begin{split} \det\left(\begin{bmatrix}1 &amp; 0 \\ 0 &amp; 0\end{bmatrix}\right) = (1 \cdot 0) - (0 \cdot 0) = 0.\end{split}\]</div>
</section>
<section id="existence-and-uniqueness">
<h2>Existence and Uniqueness<a class="headerlink" href="#existence-and-uniqueness" title="Permalink to this heading">#</a></h2>
<p>Notice that some of these transformations map multiple inputs to the same output, and some are incapable of generating certain outputs.</p>
<p>For example, the <strong>projections</strong> above can send multiple different points to the same point.</p>
<p>We need some terminology to understand these properties of linear transformations.</p>
<p><strong>Definition.</strong> A mapping <span class="math notranslate nohighlight">\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\)</span> is said to be <strong>onto</strong> <span class="math notranslate nohighlight">\(\mathbb{R}^m\)</span> if each <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> in <span class="math notranslate nohighlight">\(\mathbb{R}^m\)</span> is the image of <em>at least one</em> <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> in <span class="math notranslate nohighlight">\(\mathbb{R}^n\)</span>.</p>
<p>Informally, <span class="math notranslate nohighlight">\(T\)</span> is onto if every element of its codomain is in its range.</p>
<p>Another (important) way of thinking about this is that <span class="math notranslate nohighlight">\(T\)</span> is onto if there is a solution <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> of</p>
<div class="math notranslate nohighlight">
\[T(\mathbf{x}) = \mathbf{b}\]</div>
<p>for all possible <span class="math notranslate nohighlight">\(\mathbf{b}.\)</span></p>
<p>This is asking an <strong>existence</strong> question about a solution of the equation <span class="math notranslate nohighlight">\(T(\mathbf{x}) = \mathbf{b}\)</span> for all <span class="math notranslate nohighlight">\(\mathbf{b}.\)</span></p>
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</div>
<p>Here, we see that <span class="math notranslate nohighlight">\(T\)</span> maps points in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span> to a plane lying <strong>within</strong> <span class="math notranslate nohighlight">\(\mathbb{R}^3\)</span>.</p>
<p>That is, the range of <span class="math notranslate nohighlight">\(T\)</span> is a strict subset of the codomain of <span class="math notranslate nohighlight">\(T\)</span>.</p>
<p>So <span class="math notranslate nohighlight">\(T\)</span> is <strong>not onto</strong> <span class="math notranslate nohighlight">\(\mathbb{R}^3\)</span>.</p>
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<img alt="_images/d9503ef508361d8a203f9d0f5b3cbad26bb9c8ed5c64cabf52a89e9b33b60705.png" src="_images/d9503ef508361d8a203f9d0f5b3cbad26bb9c8ed5c64cabf52a89e9b33b60705.png" />
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</div>
<p>In this case, for every point in <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>, there is an <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> that maps to that point.</p>
<p>So, the range of <span class="math notranslate nohighlight">\(T\)</span> is equal to the codomain of <span class="math notranslate nohighlight">\(T\)</span>.</p>
<p>So <span class="math notranslate nohighlight">\(T\)</span> is <strong>onto</strong> <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>.</p>
<p>Here is an example of the reflection transformation.  The red points are the images of the blue points.</p>
<p>What about this transformation?   Is it onto <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>?</p>
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<img alt="_images/011013b27f197e33f726d222fca48672e3a089de8bc715ba2f3e70eefc004731.png" src="_images/011013b27f197e33f726d222fca48672e3a089de8bc715ba2f3e70eefc004731.png" />
</div>
</div>
<p>Here is an example of the projection transformation.  The red points (which all lie on the <span class="math notranslate nohighlight">\(x\)</span>-axis) are the images of the blue points.</p>
<p>What about this transformation?  Is it onto <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span>?</p>
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<img alt="_images/34f2bbf111de20aa749d02e3fb64a5e9d61592ae4631ead817a4d4f7d74da80c.png" src="_images/34f2bbf111de20aa749d02e3fb64a5e9d61592ae4631ead817a4d4f7d74da80c.png" />
</div>
</div>
<p><strong>Definition.</strong>  A mapping <span class="math notranslate nohighlight">\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\)</span> is said to be <strong>one-to-one</strong> if each <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> in <span class="math notranslate nohighlight">\(\mathbb{R}^m\)</span> is the image of <em>at most one</em> <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> in <span class="math notranslate nohighlight">\(\mathbb{R}^n\)</span>.</p>
<p>If <span class="math notranslate nohighlight">\(T\)</span> is one-to-one, then for each <span class="math notranslate nohighlight">\(\mathbf{b},\)</span> the equation <span class="math notranslate nohighlight">\(T(\mathbf{x}) = \mathbf{b}\)</span> has either a unique solution, or none at all.</p>
<p>This is asking a <strong>uniqueness</strong> question about a solution of the equation <span class="math notranslate nohighlight">\(T(\mathbf{x}) = \mathbf{b}\)</span> for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>.</p>
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</div>
</div>
<p>Let’s examine the relationship between these ideas and some previous definitions.</p>
<p>If for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, <span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> is consistent, is <span class="math notranslate nohighlight">\(T(\mathbf{x}) = A\mathbf{x}\)</span>  onto?  one-to-one?</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(T(\mathbf{x})\)</span> is onto.   <span class="math notranslate nohighlight">\(T(\mathbf{x})\)</span> may or may not be one-to-one.  If the system has multiple solutions for some <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, <span class="math notranslate nohighlight">\(T(\mathbf{x})\)</span> is not one-to-one.</p></li>
</ul>
<p>If for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, <span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> is consistent and has a unique solution, is <span class="math notranslate nohighlight">\(T(\mathbf{x}) = A\mathbf{x}\)</span> onto?  one-to-one?</p>
<ul class="simple">
<li><p>Yes to both.</p></li>
</ul>
<p>If it is not the case that for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, <span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> is consistent, is <span class="math notranslate nohighlight">\(T(\mathbf{x}) = A\mathbf{x}\)</span>  onto?  one-to-one?</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(T(\mathbf{x})\)</span> is <strong>not</strong> onto.   <span class="math notranslate nohighlight">\(T(\mathbf{x})\)</span> may or may not be one-to-one.</p></li>
</ul>
<p>If <span class="math notranslate nohighlight">\(T(\mathbf{x}) = A\mathbf{x}\)</span> is onto, is <span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> consistent for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>?   is the solution unique for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>?</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> is consistent for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>.  The solution may not be unique for any  <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>.</p></li>
</ul>
<p>If <span class="math notranslate nohighlight">\(T(\mathbf{x}) = A\mathbf{x}\)</span> is one-to-one, is <span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> consistent for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>?   is the solution unique for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>?</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(A\mathbf{x} = \mathbf{b}\)</span> may or may not be consistent for all <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>.  For any <span class="math notranslate nohighlight">\(\mathbf{b}\)</span>, if there is a solution, it is unique.</p></li>
</ul>
</section>
</section>

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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#geometric-linear-transformations-of-mathbb-r-2">Geometric Linear Transformations of <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span></a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#area-is-scaled-by-the-determinant">Area is Scaled by the Determinant</a></li>
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