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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2024-10-21 Mon 09:57 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>More Practice Problems</title>
<meta name="author" content="Nathan Mull" />
<meta name="generator" content="Org Mode" />
<link rel="stylesheet" type="text/css" href="myStyle.css" />
<script>
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<body>
<div id="org-div-home-and-up"><a href="material.html">↩</a></div><div id="content" class="content">
<h1 class="title">More Practice Problems</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org41f19e7">1. Intersection of Spans</a></li>
<li><a href="#orge771837">2. Dependence Relations</a></li>
<li><a href="#org787f6c0">3. True/False</a></li>
<li><a href="#orgfb1363c">4. Inverses(?)</a>
<ul>
<li><a href="#org5c25e5b">4.1. Multiply on the Right</a></li>
<li><a href="#org66491d5">4.2. Multiply on the Left</a></li>
</ul>
</li>
<li><a href="#org48aaec9">5. Linear Transformations</a></li>
</ul>
</div>
</div>
<div id="outline-container-org41f19e7" class="outline-2">
<h2 id="org41f19e7"><span class="section-number-2">1.</span> Intersection of Spans</h2>
<div class="outline-text-2" id="text-1">
<p>
Determine a vector with integer entries which appears in <i>both</i> of the following spans.
</p>
\begin{align*}
\mathsf{span}
\left\{
\begin{bmatrix}1 \\ 2 \\ -1\end{bmatrix},
\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix}
\right\}
\qquad
\mathsf{span}\left\{
\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix},
\begin{bmatrix}0 \\ 1 \\ 3\end{bmatrix}
\right\}
\end{align*}
</div>
</div>
<div id="outline-container-orge771837" class="outline-2">
<h2 id="orge771837"><span class="section-number-2">2.</span> Dependence Relations</h2>
<div class="outline-text-2" id="text-2">
<p>
Suppose that \(\{\mathbf v_1, \mathbf v_2, \mathbf v_3\}\) is a linearly
independent set of vectors and that
</p>
\begin{align*}
\mathbf u_1 &= \mathbf v_1 + \mathbf v_3 \\
\mathbf u_2 &= -2 \mathbf v_1 + \mathbf v_2 \textcolor{red}{+} \mathbf v_3 \\
\mathbf u_3 &= -3 \mathbf v_1 - \mathbf v_2 - 6 \mathbf v_3
\end{align*}
<p>
Determine a dependence relation with integer weights for the vectors
\(\{\mathbf u_1, \mathbf u_2, \mathbf u_3\}\).
</p>
</div>
</div>
<div id="outline-container-org787f6c0" class="outline-2">
<h2 id="org787f6c0"><span class="section-number-2">3.</span> True/False</h2>
<div class="outline-text-2" id="text-3">
<p>
Determine if each of the following statements are true or false. If it
is false, give a counterexample.
</p>
<ol class="org-ol">
<li>For any matrices \(A\) and \(B\), if \(AB = I\) then \(A\) is invertible
and \(B = A^{-1}\).</li>
<li>For any matrix \(A\) in \(\mathbb R^{10 \times 15}\) and any \(\mathbf
b\) in \(\mathbb R^{10}\), the matrix equation \(A\mathbf x = \mathbf b\)
has a solution.</li>
<li>For any matrices \(A\) and \(B\), if \(AB = 0\), then \(A = 0\) or \(B = 0\).</li>
<li>For any vectors \(\mathbf v_1\), \(\mathbf v_2\), \(\mathbf v_3\), if
\(\mathbf v_1 \in \mathsf{span}\{\mathbf v_2, \mathbf v_3\}\) then
\(\{ \mathbf v_1 + \mathbf v_2, \mathbf v_1 + \mathbf v_3\}\) is a
linearly dependent set.</li>
<li>For any matrices \(A\) and \(B\), if \(AB = BA\), then \(A = B\).</li>
</ol>
</div>
</div>
<div id="outline-container-orgfb1363c" class="outline-2">
<h2 id="orgfb1363c"><span class="section-number-2">4.</span> Inverses(?)</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org5c25e5b" class="outline-3">
<h3 id="org5c25e5b"><span class="section-number-3">4.1.</span> Multiply on the Right</h3>
<div class="outline-text-3" id="text-4-1">
<p>
Determine a matrix \(B\) with integer entries such that the following
equality holds.
</p>
\begin{align*}
\begin{bmatrix}
\textcolor{red}{1} & -1 & 2 \\
-3 & 4 & 2
\end{bmatrix}
B = I
\end{align*}
</div>
</div>
<div id="outline-container-org66491d5" class="outline-3">
<h3 id="org66491d5"><span class="section-number-3">4.2.</span> Multiply on the Left</h3>
<div class="outline-text-3" id="text-4-2">
<p>
Explain why it is not possible to determine a matrix \(B\) such that the
following equality holds.
</p>
\begin{align*}
B
\begin{bmatrix}
\textcolor{red}{1} & -1 & 2 \\
-3 & 4 & 2
\end{bmatrix}
= I
\end{align*}
</div>
</div>
</div>
<div id="outline-container-org48aaec9" class="outline-2">
<h2 id="org48aaec9"><span class="section-number-2">5.</span> Linear Transformations</h2>
<div class="outline-text-2" id="text-5">
<p>
Suppose that \(T : \mathbb R^3 \to \mathbb R^3\) is the linear
transformation which reflects vectors across the \(xy\) plane (i.e.,
across the plane given by the linear equation \(z = 0\)) and that \(S :
\mathbb R^3 \to \mathbb R^3\) the transformation which rotates vectors
around \(\mathsf{span}\{[1 \ \ 1 \ \ 0]^T\}\) by \(180\) degrees. Determine
the matrix which implements \(S \circ T\), the composition of \(S\) and
\(T\) (recall that \((S \circ T)(\mathbf v) = S(T(\mathbf v))\)).
</p>
</div>
</div>
</div>
</body>
</html>