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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="A2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">3.2</span> <span class="title">Standard Matrices (A2)</span>
</h2>
<article class="remark remark-like" id="remark-12"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">3.2.1</span><span class="period">.</span>
</h6>
<p id="p-499">Recall that a linear map \(T:V\rightarrow W\) satisfies</p>
<ol class="decimal">
<li id="li-335"><p id="p-500">\(T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})\) for any \(\vec{v},\vec{w} \in V\text{.}\)</p></li>
<li id="li-336"><p id="p-501">\(T(c\vec{v}) = cT(\vec{v})\) for any \(c \in \IR,\vec{v} \in V\text{.}\)</p></li>
</ol>
<p id="p-502">In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.</p></article><article class="activity project-like" id="activity-71"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.1</span><span class="period">.</span>
</h6>
<p id="p-503">Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) Compute \(T\left(\left[\begin{array}{c} 3 \\ 0 \\ 0 \end{array}\right]\right)\text{.}\)</p>
<ol class="lower-alpha">
<li id="li-337">\(\displaystyle \left[\begin{array}{c} 6 \\ 3\end{array}\right]\)</li>
<li id="li-338">\(\displaystyle \left[\begin{array}{c} -9 \\ 6 \end{array}\right]\)</li>
<li id="li-339">\(\displaystyle \left[\begin{array}{c} -4 \\ -2 \end{array}\right]\)</li>
<li id="li-340">\(\displaystyle \left[\begin{array}{c} 6 \\ -4 \end{array}\right]\)</li>
</ol></article><article class="activity project-like" id="activity-72"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.2</span><span class="period">.</span>
</h6>
<p id="p-504">Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) Compute \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right]\right)\text{.}\)</p>
<ol class="lower-alpha">
<li id="li-341"><p id="p-505">\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)</p></li>
<li id="li-342"><p id="p-506">\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)</p></li>
<li id="li-343"><p id="p-507">\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)</p></li>
<li id="li-344"><p id="p-508">\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)</p></li>
</ol></article><article class="activity project-like" id="activity-73"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.3</span><span class="period">.</span>
</h6>
<p id="p-509">Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) Compute \(T\left(\left[\begin{array}{c} -2 \\ 0 \\ -3 \end{array}\right]\right)\text{.}\)</p>
<ol class="lower-alpha">
<li id="li-345"><p id="p-510">\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)</p></li>
<li id="li-346"><p id="p-511">\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)</p></li>
<li id="li-347"><p id="p-512">\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)</p></li>
<li id="li-348"><p id="p-513">\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)</p></li>
</ol></article><article class="activity project-like" id="activity-74"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.4</span><span class="period">.</span>
</h6>
<p id="p-514">Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What piece of information would help you compute \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-349"><p id="p-515">The value of \(T\left(\left[\begin{array}{c} 0\\-4\\0\end{array}\right]\right)\text{.}\)</p></li>
<li id="li-350"><p id="p-516">The value of \(T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)\text{.}\)</p></li>
<li id="li-351"><p id="p-517">The value of \(T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)\text{.}\)</p></li>
<li id="li-352"><p id="p-518">Any of the above.</p></li>
</ol></article><article class="fact theorem-like" id="fact-15"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">3.2.2</span><span class="period">.</span>
</h6>
<p id="p-519">Consider any basis \(\{\vec b_1,\dots,\vec b_n\}\) for \(V\text{.}\) Since every vector \(\vec v\) can be written as a linear combination of basis vectors, \(x_1\vec b_1+\dots+ x_n\vec b_n\text{,}\) we may compute \(T(\vec v)\) as follows:</p>
<div class="displaymath">
\begin{equation*}
T(\vec v)=T(x_1\vec b_1+\dots+ x_n\vec b_n)=
x_1T(\vec b_1)+\dots+x_nT(\vec b_n)
.
\end{equation*}
</div>
<p id="p-520">Therefore any linear transformation \(T:V \rightarrow W\) can be defined by just describing the values of \(T(\vec b_i)\text{.}\)</p>
<p id="p-521">Put another way, the images of the basis vectors <dfn class="terminology">determine</dfn> the transformation \(T\text{.}\)</p></article><article class="definition definition-like" id="definition-19"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.2.3</span><span class="period">.</span>
</h6>
<p id="p-522">Since linear transformation \(T:\IR^n\to\IR^m\) is determined by the standard basis \(\{\vec e_1,\dots,\vec e_n\}\text{,}\) it's convenient to store this information in the \(m\times n\) <dfn class="terminology">standard matrix</dfn> \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\text{.}\)</p>
<p id="p-523">For example, let \(T: \IR^3 \rightarrow \IR^2\) be the linear map determined by the following values for \(T\) applied to the standard basis of \(\IR^3\text{.}\)</p>
<div class="displaymath">
\begin{equation*}
\scriptsize
T\left(\vec e_1 \right)
=
T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 3 \\ 2\end{array}\right]
\hspace{2em}
T\left(\vec e_2 \right)
=
T\left(\left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} -1 \\ 4\end{array}\right]
\hspace{2em}
T\left(\vec e_3 \right)
=
T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} 5 \\ 0\end{array}\right]
\end{equation*}
</div>
<p id="p-524">Then the standard matrix corresponding to \(T\) is</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc}T(\vec e_1) & T(\vec e_2) & T(\vec e_3)\end{array}\right]
=
\left[\begin{array}{ccc}3 & -1 & 5 \\ 2 & 4 & 0 \end{array}\right]
.
\end{equation*}
</div></article><article class="activity project-like" id="activity-75"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.5</span><span class="period">.</span>
</h6>
<p id="p-525">Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\vec e_1 \right)
=
\left[\begin{array}{c} 0 \\ 3 \\ -2\end{array}\right]
\hspace{2em}
T\left(\vec e_2 \right)
=
\left[\begin{array}{c} -3 \\ 0 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_3 \right)
=
\left[\begin{array}{c} 4 \\ -2 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_4 \right)
=
\left[\begin{array}{c} 2 \\ 0 \\ 0\end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Write the standard matrix \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\) for \(T\text{.}\)</p></article><article class="activity project-like" id="activity-76"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.6</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-26">
<p id="p-526">Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x+3z \\ 2x-y-4z \end{array}\right]
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-61"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-527">Compute \(T(\vec e_1)\text{,}\) \(T(\vec e_2)\text{,}\) and \(T(\vec e_3)\text{.}\)</p></article><article class="task exercise-like" id="task-62"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-528">Find the standard matrix for \(T\text{.}\)</p></article></article><article class="fact theorem-like" id="fact-16"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">3.2.4</span><span class="period">.</span>
</h6>
<p id="p-529">Because every linear map \(T:\IR^m\to\IR^n\) has a linear combination of the variables in each component, and thus \(T(\vec e_i)\) yields exactly the coefficients of \(x_i\text{,}\) the standard matrix for \(T\) is simply an ordered list of the coefficients of the \(x_i\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c}x\\y\\z\\w\end{array}\right]\right)
=
\left[\begin{array}{c}
ax+by+cz+dw \\
ex+fy+gz+hw
\end{array}\right]
\hspace{2em}
A
=
\left[\begin{array}{cccc}
a & b & c & d \\
e & f & g & h
\end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-77"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.7</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-27">
<p id="p-530">Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} 3 & -2 & -1 \\ 4 & 5 & 2 \\ 0 & -2 & 1 \end{array}\right]
.
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-63"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-531">Compute \(T\left(\left[\begin{array}{c} 1\\ 2 \\ 3 \end{array}\right] \right) \text{.}\)</p></article><article class="task exercise-like" id="task-64"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-532">Compute \(T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) \text{.}\)</p></article></article><article class="activity project-like" id="activity-78"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.2.8</span><span class="period">.</span>
</h6>
<p id="p-533">Compute the following linear transformations of vectors given their standard matrices.</p>
<div class="displaymath">
\begin{equation*}
T_1\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)
\text{ for the standard matrix }
A_1=\left[\begin{array}{cc}4&3\\0&-1\\1&1\\3&0\end{array}\right]
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
T_2\left(\left[\begin{array}{c}1\\1\\0\\-3\end{array}\right]\right)
\text{ for the standard matrix }
A_2=\left[\begin{array}{cccc}4&3&0&-1\\1&1&3&0\end{array}\right]
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
T_3\left(\left[\begin{array}{c}0\\-2\\0\end{array}\right]\right)
\text{ for the standard matrix }
A_3=\left[\begin{array}{ccc}4&3&0\\0&-1&3\\5&1&1\\3&0&0\end{array}\right]
\end{equation*}
</div></article><section class="exercises" id="exercises-14"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">3.2.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-66"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-353">Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 4 \) given by<div class="displaymath">
\begin{equation*}
S\left( \left[\begin{array}{c}
x \\
y
\end{array}\right] \right) = \left[\begin{array}{c}
x - 5 \, y \\
-x + 6 \, y \\
0 \\
x - 2 \, y
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-354">Let \(T:\mathbb{R}^ 2 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc}
-1 & 0 \\
-1 & -4 \\
2 & 5 \\
-1 & 0
\end{array}\right] .
\end{equation*}
</div>Compute \(T\left( \left[\begin{array}{c}
-2 \\
8
\end{array}\right] \right)\text{.}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-66" id="answer-66"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-66"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-355"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc}
1 & -5 \\
-1 & 6 \\
0 & 0 \\
1 & -2
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-356"><div class="displaymath">
\begin{equation*}
T\left( \left[\begin{array}{c}
-2 \\
8
\end{array}\right] \right)= \left[\begin{array}{c}
2 \\
-30 \\
36 \\
2
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-67"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-357">Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) given by<div class="displaymath">
\begin{equation*}
S\left( \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \right) = \left[\begin{array}{c}
-x + 3 \, y + 4 \, z \\
-x - z \\
2 \, x - 2 \, y - z \\
-3 \, x + y - 6 \, z
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-358">Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) be the linear transformation given by the standard matrix<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
2 & 6 & 1 & -3 \\
1 & 3 & 1 & -2
\end{array}\right] .
\end{equation*}
</div>Compute \(T\left( \left[\begin{array}{c}
-3 \\
-7 \\
6 \\
-5
\end{array}\right] \right)\text{.}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-67" id="answer-67"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-67"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-359"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc}
-1 & 3 & 4 \\
-1 & 0 & -1 \\
2 & -2 & -1 \\
-3 & 1 & -6
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-360"><div class="displaymath">
\begin{equation*}
T\left( \left[\begin{array}{c}
-3 \\
-7 \\
6 \\
-5
\end{array}\right] \right)= \left[\begin{array}{c}
-27 \\
-8
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-68"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-361">Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) given by<div class="displaymath">
\begin{equation*}
S\left( \left[\begin{array}{c}
x \\
y \\
z \\
{w}
\end{array}\right] \right) = \left[\begin{array}{c}
-2 \, x + y - 2 \, {w} \\
x - y
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-362">Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 1 \) be the linear transformation given by the standard matrix<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
1 & -1 & -2 & -1
\end{array}\right] .
\end{equation*}
</div>Compute \(T\left( \left[\begin{array}{c}
6 \\
-3 \\
5 \\
-7
\end{array}\right] \right)\text{.}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-68" id="answer-68"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-68"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-363"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
-2 & 1 & 0 & -2 \\
1 & -1 & 0 & 0
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-364"><div class="displaymath">
\begin{equation*}
T\left( \left[\begin{array}{c}
6 \\
-3 \\
5 \\
-7
\end{array}\right] \right)= \left[\begin{array}{c}
6
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-69"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-365">Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 3 \) given by<div class="displaymath">
\begin{equation*}
S\left( \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \right) = \left[\begin{array}{c}
x \\
y - 4 \, z \\
-y + 5 \, z
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-366">Let \(T:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc}
-5 & 3 \\
3 & -2 \\
0 & 2
\end{array}\right] .
\end{equation*}
</div>Compute \(T\left( \left[\begin{array}{c}
-5 \\
3
\end{array}\right] \right)\text{.}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-69" id="answer-69"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-69"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-367"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & -4 \\
0 & -1 & 5
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-368"><div class="displaymath">
\begin{equation*}
T\left( \left[\begin{array}{c}
-5 \\
3
\end{array}\right] \right)= \left[\begin{array}{c}
34 \\
-21 \\
6
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-70"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<ol class="lower-alpha">
<li id="li-369">Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) given by<div class="displaymath">
\begin{equation*}
S\left( \left[\begin{array}{c}
x \\
y \\
z \\
{w}
\end{array}\right] \right) = \left[\begin{array}{c}
2 \, x - 3 \, y + z + 6 \, {w} \\
x - y + 2 \, {w}
\end{array}\right] .
\end{equation*}
</div>
</li>
<li id="li-370">Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
3 & 0 & 4 & 5 \\
0 & 1 & 1 & 3 \\
-1 & -1 & -2 & -4
\end{array}\right] .
\end{equation*}
</div>Compute \(T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right] \right)\text{.}\)</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-70" id="answer-70"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-70"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-371"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
2 & -3 & 1 & 6 \\
1 & -1 & 0 & 2
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-372"><div class="displaymath">
\begin{equation*}
T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right] \right)= \left[\begin{array}{c}
33 \\
9 \\
-19
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><p id="p-534"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/A2/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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