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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="V8"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.8</span> <span class="title">Polynomial and Matrix Spaces (V8)</span>
</h2>
<article class="fact theorem-like" id="fact-11"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.8.1</span><span class="period">.</span>
</h6>
<p id="p-396">Every vector space with finite dimension, that is, every vector space \(V\) with a basis of the form \(\{\vec v_1,\vec v_2,\dots,\vec v_n\}\) is said to be <dfn class="terminology">isomorphic</dfn> to a Euclidean space \(\IR^n\text{,}\) since there exists a natural correspondance between vectors in \(V\) and vectors in \(\IR^n\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
c_1\vec v_1+c_2\vec v_2+\dots+c_n\vec v_n
\leftrightarrow
\left[\begin{array}{c}
c_1\\c_2\\\vdots\\c_n
\end{array}\right]
\end{equation*}
</div></article><article class="observation remark-like" id="observation-13"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.8.2</span><span class="period">.</span>
</h6>
<p id="p-397">We've already been taking advantage of the previous fact by converting polynomials and matrices into Euclidean vectors. Since \(\P^3\) and \(M_{2,2}\) are both four-dimensional:</p>
<div class="displaymath">
\begin{equation*}
4x^3+0x^2-1x+5
\leftrightarrow
\left[\begin{array}{c}
4\\0\\-1\\5
\end{array}\right]
\leftrightarrow
\left[\begin{array}{cc}
4&0\\-1&5
\end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-60"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.8.1</span><span class="period">.</span>
</h6>
<p id="p-398">Suppose \(W\) is a subspace of \(\P^8\text{,}\) and you know that the set \(\{ x^3+x, x^2+1, x^4-x \}\) is a linearly independent subset of \(W\text{.}\) What can you conclude about \(W\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-276"><p id="p-399">The dimension of \(W\) is 3 or less.</p></li>
<li id="li-277"><p id="p-400">The dimension of \(W\) is exactly 3.</p></li>
<li id="li-278"><p id="p-401">The dimension of \(W\) is 3 or more.</p></li>
</ol></article><article class="activity project-like" id="activity-61"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.8.2</span><span class="period">.</span>
</h6>
<p id="p-402">Suppose \(W\) is a subspace of \(\P^8\text{,}\) and you know that \(W\) is spanned by the six vectors</p>
<div class="displaymath">
\begin{equation*}
\{ x^4-x,x^3+x,x^3+x+1,x^4+2x,x^3,2x+1\}.
\end{equation*}
</div>
<p data-braille="continuation">What can you conclude about \(W\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-279"><p id="p-403">The dimension of \(W\) is 6 or less.</p></li>
<li id="li-280"><p id="p-404">The dimension of \(W\) is exactly 3.</p></li>
<li id="li-281"><p id="p-405">The dimension of \(W\) is 6 or more.</p></li>
</ol></article><article class="observation remark-like" id="observation-14"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.8.3</span><span class="period">.</span>
</h6>
<p id="p-406">The space of polynomials \(\P\) (of <em class="emphasis">any</em> degree) has the basis \(\{1,x,x^2,x^3,\dots\}\text{,}\) so it is a natural example of an infinite-dimensional vector space.</p>
<p id="p-407">Since \(\P\) and other infinite-dimensional spaces cannot be treated as an isomorphic finite-dimensional Euclidean space \(\IR^n\text{,}\) vectors in such spaces cannot be studied by converting them into Euclidean vectors. Fortunately, most of the examples we will be interested in for this course will be finite-dimensional.</p></article><section class="exercises" id="exercises-11"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.8.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-51"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-408">Consider the statement <article class="claim theorem-like" id="claim-61"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.4</span><span class="period">.</span>
</h6>\(\left\{ -4 \, x^{3} + 2 \, x^{2} - x - 6 , -5 \, x^{3} + 3 \, x^{2} + 5 \, x + 2 , -5 \, x^{3} + 3 \, x^{2} + x - 4 , 10 \, x^{3} - 6 \, x^{2} - 10 \, x - 4 \right\} \)</article></p>
<ol class="lower-alpha">
<li id="li-282">Write an equivalent statement using a polynomial equation.</li>
<li id="li-283">Explain why your statement is true or false.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-51" id="answer-51"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-51"><div class="answer solution-like">
<div class="displaymath" id="p-409">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
-6 & 2 & -4 & -4 \\
-1 & 5 & 1 & -10 \\
2 & 3 & 3 & -6 \\
-4 & -5 & -5 & 10
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -2 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-284">The statement <article class="claim theorem-like" id="claim-62"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.5</span><span class="period">.</span>
</h6>\(\left\{ -4 \, x^{3} + 2 \, x^{2} - x - 6 , -5 \, x^{3} + 3 \, x^{2} + 5 \, x + 2 , -5 \, x^{3} + 3 \, x^{2} + x - 4 , 10 \, x^{3} - 6 \, x^{2} - 10 \, x - 4 \right\} \)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-63"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.6</span><span class="period">.</span>
</h6>
<div class="displaymath">
\begin{equation*}
y_{1} \left( -4 \, x^{3} + 2 \, x^{2} - x - 6 \right) + y_{2} \left( -5 \, x^{3} + 3 \, x^{2} + 5 \, x + 2 \right) + y_{3} \left( -5 \, x^{3} + 3 \, x^{2} + x - 4 \right) + y_{4} \left( 10 \, x^{3} - 6 \, x^{2} - 10 \, x - 4 \right) = 0
\end{equation*}
</div></article>
</li>
<li id="li-285">The set of polynomials \(\left\{ -4 \, x^{3} + 2 \, x^{2} - x - 6 , -5 \, x^{3} + 3 \, x^{2} + 5 \, x + 2 , -5 \, x^{3} + 3 \, x^{2} + x - 4 , 10 \, x^{3} - 6 \, x^{2} - 10 \, x - 4 \right\} \)is linearly dependent.</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-52"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-410">Consider the statement <article class="claim theorem-like" id="claim-64"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.7</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
0 & 1 \\
-6 & 0
\end{array}\right] , \left[\begin{array}{cc}
5 & 3 \\
-5 & 5
\end{array}\right] , \left[\begin{array}{cc}
-4 & 3 \\
-5 & -2
\end{array}\right] , \left[\begin{array}{cc}
-3 & -4 \\
1 & -4
\end{array}\right] \right\} \)</article></p>
<ol class="lower-alpha">
<li id="li-286">Write an equivalent statement using a matrix equation.</li>
<li id="li-287">Explain why your statement is true or false.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-52" id="answer-52"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-52"><div class="answer solution-like">
<div class="displaymath" id="p-411">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
0 & 5 & -4 & -3 \\
1 & 3 & 3 & -4 \\
-6 & -5 & -5 & 1 \\
0 & 5 & -2 & -4
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-288">The statement <article class="claim theorem-like" id="claim-65"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.8</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
0 & 1 \\
-6 & 0
\end{array}\right] , \left[\begin{array}{cc}
5 & 3 \\
-5 & 5
\end{array}\right] , \left[\begin{array}{cc}
-4 & 3 \\
-5 & -2
\end{array}\right] , \left[\begin{array}{cc}
-3 & -4 \\
1 & -4
\end{array}\right] \right\} \)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-66"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.9</span><span class="period">.</span>
</h6>
<div class="displaymath">
\begin{equation*}
y_{1} \left[\begin{array}{cc}
0 & 1 \\
-6 & 0
\end{array}\right] + y_{2} \left[\begin{array}{cc}
5 & 3 \\
-5 & 5
\end{array}\right] + y_{3} \left[\begin{array}{cc}
-4 & 3 \\
-5 & -2
\end{array}\right] + y_{4} \left[\begin{array}{cc}
-3 & -4 \\
1 & -4
\end{array}\right] = \left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]
\end{equation*}
</div></article>
</li>
<li id="li-289">The set of matrices \(\left\{ \left[\begin{array}{cc}
0 & 1 \\
-6 & 0
\end{array}\right] , \left[\begin{array}{cc}
5 & 3 \\
-5 & 5
\end{array}\right] , \left[\begin{array}{cc}
-4 & 3 \\
-5 & -2
\end{array}\right] , \left[\begin{array}{cc}
-3 & -4 \\
1 & -4
\end{array}\right] \right\} \)is linearly independent.</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-53"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-412">Consider the statement <article class="claim theorem-like" id="claim-67"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.10</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
-4 & 0 \\
3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-2 & -1
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & 2
\end{array}\right] , \left[\begin{array}{cc}
17 & -7 \\
0 & 8
\end{array}\right] \right\} \)</article></p>
<ol class="lower-alpha">
<li id="li-290">Write an equivalent statement using a matrix equation.</li>
<li id="li-291">Explain why your statement is true or false.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-53" id="answer-53"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-53"><div class="answer solution-like">
<div class="displaymath" id="p-413">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
-4 & -4 & 4 & 17 \\
0 & 4 & 0 & -7 \\
3 & -2 & -3 & 0 \\
-2 & -1 & 2 & 8
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-292">The statement <article class="claim theorem-like" id="claim-68"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.11</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
-4 & 0 \\
3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-2 & -1
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & 2
\end{array}\right] , \left[\begin{array}{cc}
17 & -7 \\
0 & 8
\end{array}\right] \right\} \)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-69"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.12</span><span class="period">.</span>
</h6>
<div class="displaymath">
\begin{equation*}
y_{1} \left[\begin{array}{cc}
-4 & 0 \\
3 & -2
\end{array}\right] + y_{2} \left[\begin{array}{cc}
-4 & 4 \\
-2 & -1
\end{array}\right] + y_{3} \left[\begin{array}{cc}
4 & 0 \\
-3 & 2
\end{array}\right] + y_{4} \left[\begin{array}{cc}
17 & -7 \\
0 & 8
\end{array}\right] = \left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]
\end{equation*}
</div></article>
</li>
<li id="li-293">The set of matrices \(\left\{ \left[\begin{array}{cc}
-4 & 0 \\
3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-2 & -1
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & 2
\end{array}\right] , \left[\begin{array}{cc}
17 & -7 \\
0 & 8
\end{array}\right] \right\} \)is linearly dependent.</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-54"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-414">Consider the statement <article class="claim theorem-like" id="claim-70"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.13</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
2 & 0 \\
-4 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-4 & -4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-3 & 0 \\
2 & -5
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-2 & 3 \\
2 & -1
\end{array}\right] \right\} \)\(\mathrm{M}_{2,2}\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-294">Write an equivalent statement using a matrix equation.</li>
<li id="li-295">Explain why your statement is true or false.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-54" id="answer-54"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-54"><div class="answer solution-like">
<div class="displaymath" id="p-415">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccccc}
2 & -4 & -4 & -3 & 4 & -2 \\
0 & 4 & -4 & 0 & 0 & 3 \\
-4 & -1 & -1 & 2 & -3 & 2 \\
-2 & -3 & -3 & -5 & -2 & -1
\end{array}\right] = \left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & \frac{9}{8} & -\frac{3}{8} \\
0 & 1 & 0 & 0 & -\frac{4}{11} & \frac{37}{88} \\
0 & 0 & 1 & 0 & -\frac{4}{11} & -\frac{29}{88} \\
0 & 0 & 0 & 1 & \frac{17}{44} & \frac{13}{44}
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-296">The statement <article class="claim theorem-like" id="claim-71"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.14</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
2 & 0 \\
-4 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-4 & -4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-3 & 0 \\
2 & -5
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-2 & 3 \\
2 & -1
\end{array}\right] \right\} \)\(\mathrm{M}_{2,2}\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-72"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.15</span><span class="period">.</span>
</h6>
<div class="displaymath">
\begin{equation*}
y_{1} \left[\begin{array}{cc}
2 & 0 \\
-4 & -2
\end{array}\right] + y_{2} \left[\begin{array}{cc}
-4 & 4 \\
-1 & -3
\end{array}\right] + y_{3} \left[\begin{array}{cc}
-4 & -4 \\
-1 & -3
\end{array}\right] + y_{4} \left[\begin{array}{cc}
-3 & 0 \\
2 & -5
\end{array}\right] + y_{5} \left[\begin{array}{cc}
4 & 0 \\
-3 & -2
\end{array}\right] + y_{6} \left[\begin{array}{cc}
-2 & 3 \\
2 & -1
\end{array}\right] =B
\end{equation*}
</div>\(B \in \mathrm{M}_{2,2}\text{.}\)</article>
</li>
<li id="li-297">The set of matrices \(\left\{ \left[\begin{array}{cc}
2 & 0 \\
-4 & -2
\end{array}\right] , \left[\begin{array}{cc}
-4 & 4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-4 & -4 \\
-1 & -3
\end{array}\right] , \left[\begin{array}{cc}
-3 & 0 \\
2 & -5
\end{array}\right] , \left[\begin{array}{cc}
4 & 0 \\
-3 & -2
\end{array}\right] , \left[\begin{array}{cc}
-2 & 3 \\
2 & -1
\end{array}\right] \right\} \) spans \(\mathrm{M}_{2,2}\text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-55"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-416">Consider the statement <article class="claim theorem-like" id="claim-73"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.16</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
3 & 4 \\
-3 & -5
\end{array}\right] , \left[\begin{array}{cc}
0 & 3 \\
-2 & -2
\end{array}\right] , \left[\begin{array}{cc}
-5 & -2 \\
-5 & 1
\end{array}\right] , \left[\begin{array}{cc}
6 & 8 \\
-6 & -10
\end{array}\right] \right\} \)\(\mathrm{M}_{2,2}\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-298">Write an equivalent statement using a matrix equation.</li>
<li id="li-299">Explain why your statement is true or false.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-55" id="answer-55"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-55"><div class="answer solution-like">
<div class="displaymath" id="p-417">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
3 & 0 & -5 & 6 \\
4 & 3 & -2 & 8 \\
-3 & -2 & -5 & -6 \\
-5 & -2 & 1 & -10
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-300">The statement <article class="claim theorem-like" id="claim-74"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.17</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{cc}
3 & 4 \\
-3 & -5
\end{array}\right] , \left[\begin{array}{cc}
0 & 3 \\
-2 & -2
\end{array}\right] , \left[\begin{array}{cc}
-5 & -2 \\
-5 & 1
\end{array}\right] , \left[\begin{array}{cc}
6 & 8 \\
-6 & -10
\end{array}\right] \right\} \)\(\mathrm{M}_{2,2}\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-75"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.8.18</span><span class="period">.</span>
</h6>
<div class="displaymath">
\begin{equation*}
y_{1} \left[\begin{array}{cc}
3 & 4 \\
-3 & -5
\end{array}\right] + y_{2} \left[\begin{array}{cc}
0 & 3 \\
-2 & -2
\end{array}\right] + y_{3} \left[\begin{array}{cc}
-5 & -2 \\
-5 & 1
\end{array}\right] + y_{4} \left[\begin{array}{cc}
6 & 8 \\
-6 & -10
\end{array}\right] =B
\end{equation*}
</div>\(B \in \mathrm{M}_{2,2}\text{.}\)</article>
</li>
<li id="li-301">The set of matrices \(\left\{ \left[\begin{array}{cc}
3 & 4 \\
-3 & -5
\end{array}\right] , \left[\begin{array}{cc}
0 & 3 \\
-2 & -2
\end{array}\right] , \left[\begin{array}{cc}
-5 & -2 \\
-5 & 1
\end{array}\right] , \left[\begin{array}{cc}
6 & 8 \\
-6 & -10
\end{array}\right] \right\} \) does not span \(\mathrm{M}_{2,2}\text{.}\)</li>
</ol>
</div></div>
</div></article><p id="p-418"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V8/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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