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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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<a href="E.html" data-scroll="E"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations (E)</span></a><ul>
<li><a href="E1.html" data-scroll="E1">Linear Systems, Vector Equations, and Augmented Matrices (E1)</a></li>
<li><a href="E2.html" data-scroll="E2">Row Reduction of Matrices (E2)</a></li>
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<a href="V.html" data-scroll="V"><span class="codenumber">2</span> <span class="title">Vector Spaces (V)</span></a><ul>
<li><a href="V1.html" data-scroll="V1">Vector Spaces (V1)</a></li>
<li><a href="V2.html" data-scroll="V2">Linear Combinations (V2)</a></li>
<li><a href="V3.html" data-scroll="V3">Spanning Sets (V3)</a></li>
<li><a href="V4.html" data-scroll="V4">Subspaces (V4)</a></li>
<li><a href="V5.html" data-scroll="V5">Linear Independence (V5)</a></li>
<li><a href="V6.html" data-scroll="V6">Identifying a Basis (V6)</a></li>
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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="E3"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">1.3</span> <span class="title">Solving Linear Systems (E3)</span>
</h2>
<article class="activity project-like" id="activity-14"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.1</span><span class="period">.</span>
</h6>
<p id="p-70">Free browser-based technologies for mathematical computation are available online.</p>
<ul class="disc">
<li id="li-86"><p id="p-71">Go to <a class="external" href="https://sagecell.sagemath.org/" target="_blank"><code class="code-inline tex2jax_ignore">https://sagecell.sagemath.org/</code></a>.</p></li>
<li id="li-87"><p id="p-72">Type <code class="code-inline tex2jax_ignore">A=Matrix([[1,3,4],[2,5,7]])</code> to store the matrix \(\left[\begin{array}{} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\) in the variable \(A\text{.}\) Then add the line <code class="code-inline tex2jax_ignore">show(A)</code> and use the <kbd class="kbdkey">Evaluate</kbd> button to preview it.</p></li>
<li id="li-88"><p id="p-73">Add the line <code class="code-inline tex2jax_ignore">show(A.rref())</code> and use <kbd class="kbdkey">Evaluate</kbd> to compute the reduced row echelon form of \(A\text{.}\)</p></li>
</ul>
<p id="p-74">Since the vertical bar in an augmented matrix does not affect row operations, the \(\RREF\) of \(\left[\begin{array}{cc|c} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\) may be computed in the same way.</p></article><article class="activity project-like" id="activity-15"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-3">
<p id="p-75">Consider the following system of equations.</p>
<div class="displaymath">
\begin{align*}
3x_1 &\,-\,& 2x_2 &\,+\,& 13x_3 &\,=\,& 6\\
2x_1 &\,-\,& 2x_2 &\,+\,& 10x_3 &\,=\,& 2\\
-x_1 &\,+\,& 3x_2 &\,-\,& 6x_3 &\,=\,& 11
\end{align*}
</div>
</div>
<article class="task exercise-like" id="task-5"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-76">Convert this to an augmented matrix and use technology to compute its reduced row echelon form:</p>
<div class="displaymath">
\begin{equation*}
\RREF
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
=
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
\end{equation*}
</div></article><article class="task exercise-like" id="task-6"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-77">Use the \(\RREF\) matrix to write a linear system equivalent to the original system. Then find its solution set.</p></article></article><article class="activity project-like" id="activity-16"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-4">
<p id="p-78">Consider the vector equation</p>
<div class="displaymath">
\begin{equation*}
x_1 \left[\begin{array}{c} 3 \\ 2\\ -1 \end{array}\right]
+x_2 \left[\begin{array}{c}-2 \\ -2 \\ 0 \end{array}\right]
+x_3\left[\begin{array}{c} 13 \\ 10 \\ -3 \end{array}\right]
=\left[\begin{array}{c} 6 \\ 2 \\ 1 \end{array}\right]
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-7"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-79">Convert this to an augmented matrix and use technology to compute its reduced row echelon form:</p>
<div class="displaymath">
\begin{equation*}
\RREF
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
=
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
\end{equation*}
</div></article><article class="task exercise-like" id="task-8"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-80">Use the \(\RREF\) matrix to write a linear system equivalent to the original system. Then find its solution set.</p></article></article><article class="activity project-like" id="activity-17"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.4</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-5">
<p id="p-81">Consider the following linear system.</p>
<div class="displaymath">
\begin{alignat*}{4}
x_1 &+ 2x_2 &+ 3x_3 &= 1\\
2x_1 &+ 4x_2 &+ 8x_3 &= 0
\end{alignat*}
</div>
</div>
<article class="task exercise-like" id="task-9"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-82">Find its corresponding augmented matrix \(A\) and use technology to find \(\RREF(A)\text{.}\)</p></article><article class="task exercise-like" id="task-10"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-83">How many solutions do these linear systems have?</p></article></article><article class="activity project-like" id="activity-18"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.5</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-6">
<p id="p-84">Consider the simple linear system equivalent to the system from the previous activity:</p>
<div class="displaymath">
\begin{alignat*}{3}
x_1 &+ 2x_2 & &= 4\\
& &\phantom{+}x_3 &= -1
\end{alignat*}
</div>
</div>
<article class="task exercise-like" id="task-11"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-85">Let \(x_1=a\) and write the solution set in the form \(\setBuilder
{
\left[\begin{array}{c} a \\ \unknown \\ \unknown \end{array}\right]
}{
a \in \IR
}
\text{.}\)</p></article><article class="task exercise-like" id="task-12"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-86">Let \(x_2=b\) and write the solution set in the form \(\setBuilder
{
\left[\begin{array}{c} \unknown \\ b \\ \unknown \end{array}\right]
}{
b \in \IR
}
\text{.}\)</p></article><article class="task exercise-like" id="task-13"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-87">Which of these was easier? What features of the RREF matrix \(\left[\begin{array}{ccc|c}
\circledNumber{1} & 2 & 0 & 4 \\
0 & 0 & \circledNumber{1} & -1
\end{array}\right]\) caused this?</p></article></article><article class="definition definition-like" id="definition-8"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.3.1</span><span class="period">.</span>
</h6>
<p id="p-88">Recall that the pivots of a matrix in \(\RREF\) form are the leading \(1\)s in each non-zero row.</p>
<p id="p-89">The pivot columns in an augmented matrix correspond to the <dfn class="terminology">bound variables</dfn> in the system of equations (\(x_1,x_3\) below). The remaining variables are called <dfn class="terminology">free variables</dfn> (\(x_2\) below).</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
\circledNumber{1} & 2 & 0 & 4 \\
0 & 0 & \circledNumber{1} & -1
\end{array}\right]
\end{equation*}
</div>
<p id="p-90">To efficiently solve a system in RREF form, assign letters to the free variables, and then solve for the bound variables.</p></article><article class="activity project-like" id="activity-19"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.3.6</span><span class="period">.</span>
</h6>
<p id="p-91">Find the solution set for the system</p>
<div class="displaymath">
\begin{alignat*}{6}
2x_1&\,-\,&2x_2&\,-\,&6x_3&\,+\,&x_4&\,-\,&x_5&\,=\,&3 \\
-x_1&\,+\,&x_2&\,+\,&3x_3&\,-\,&x_4&\,+\,&2x_5 &\,=\,& -3 \\
x_1&\,-\,&2x_2&\,-\,&x_3&\,+\,&x_4&\,+\,&x_5 &\,=\,& 2
\end{alignat*}
</div>
<p data-braille="continuation">by row-reducing its augmented matrix, and then assigning letters to the free variables (given by non-pivot columns) and solving for the bound variables (given by pivot columns) in the corresponding linear system.</p></article><article class="observation remark-like" id="observation-2"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">1.3.2</span><span class="period">.</span>
</h6>
<p id="p-92">The solution set to the system</p>
<div class="displaymath">
\begin{alignat*}{6}
2x_1&\,-\,&2x_2&\,-\,&6x_3&\,+\,&x_4&\,-\,&x_5&\,=\,&3 \\
-x_1&\,+\,&x_2&\,+\,&3x_3&\,-\,&x_4&\,+\,&2x_5 &\,=\,& -3 \\
x_1&\,-\,&2x_2&\,-\,&x_3&\,+\,&x_4&\,+\,&x_5 &\,=\,& 2
\end{alignat*}
</div>
<p data-braille="continuation">may be written as</p>
<div class="displaymath">
\begin{equation*}
\setBuilder
{
\left[\begin{array}{c}
1+5a+2b \\
1+2a+3b \\
a \\
3+3b \\
b
\end{array}\right]
}{
a,b\in \IR
}\text{.}
\end{equation*}
</div></article><article class="remark remark-like" id="remark-7"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.3.3</span><span class="period">.</span>
</h6>
<p id="p-93">Don't forget to correctly express the solution set of a linear system. Systems with zero or one solutions may be written by listing their elements, while systems with infinitely-many solutions may be written using set-builder notation.</p>
<ul class="disc">
<li id="li-89"><p id="p-94"><em class="emphasis">Consistent with one solution</em>: e.g. \(\setList{ \left[\begin{array}{c}1\\2\\3\end{array}\right] }\)</p></li>
<li id="li-90"><p id="p-95"><em class="emphasis">Consistent with infinitely-many solutions</em>: e.g. \(\setBuilder
{
\left[\begin{array}{c}1\\2-3a\\a\end{array}\right]
}{
a\in\IR
}\)</p></li>
<li id="li-91"><p id="p-96"><em class="emphasis">Inconsistent</em>: \(\emptyset\) or \(\{\}\)</p></li>
</ul></article><section class="exercises" id="exercises-3"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">1.3.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-11"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-97">Show how to find the solution set for the following system of linear equations.</p>
<div class="displaymath">
\begin{alignat*}{5}
x_{1} &-& 2 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 4\\
2 \, x_{1} &-& 3 \, x_{2} &-& 3 \, x_{3} &+& 5 \, x_{4} &=& 6\\
&-& 3 \, x_{2} &+& 3 \, x_{3} &+& 3 \, x_{4} &=& 6
\end{alignat*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-11" id="answer-11"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-11"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\mathrm{RREF} \left[\begin{array}{cccc|c}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -3 & 3 & 3 & 6
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & -3 & 1 & 0 \\
0 & 1 & -1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-98">The solution set is \(\left\{ \left[\begin{array}{c}
3 \, a - b \\
a + b - 2 \\
a \\
b
\end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\} \)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-12"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-99">Show how to find the solution set for the following system of linear equations.</p>
<div class="displaymath">
\begin{alignat*}{4}
x_{1} & & &-& x_{3} &=& 4\\
x_{1} &+& x_{2} & & &=& 3\\
x_{1} &+& x_{2} &+& x_{3} &=& -2\\
& & & & x_{3} &=& -4
\end{alignat*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-12" id="answer-12"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-12"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\mathrm{RREF} \left[\begin{array}{ccc|c}
1 & 0 & -1 & 4 \\
1 & 1 & 0 & 3 \\
1 & 1 & 1 & -2 \\
0 & 0 & 1 & -4
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{equation*}
</div>
<p id="p-100">The solution set is \(\left\{\right\} \)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-13"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-101">Show how to find the solution set for the following system of linear equations.</p>
<div class="displaymath">
\begin{alignat*}{4}
4 \, x_{1} &+& x_{2} &+& 7 \, x_{3} &=& 6\\
-5 \, x_{1} &-& 4 \, x_{2} &-& 6 \, x_{3} &=& -2\\
-x_{1} &+& x_{2} &-& 3 \, x_{3} &=& -4\\
-3 \, x_{1} & & &-& 6 \, x_{3} &=& -6
\end{alignat*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-13" id="answer-13"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-13"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\mathrm{RREF} \left[\begin{array}{ccc|c}
4 & 1 & 7 & 6 \\
-5 & -4 & -6 & -2 \\
-1 & 1 & -3 & -4 \\
-3 & 0 & -6 & -6
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 2 & 2 \\
0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-102">The solution set is \(\left\{ \left[\begin{array}{c}
-2 \, a + 2 \\
a - 2 \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-14"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-103">Show how to find the solution set for the following system of linear equations.</p>
<div class="displaymath">
\begin{alignat*}{4}
x_{1} &-& x_{2} &-& x_{3} &=& -6\\
& & x_{2} &-& 3 \, x_{3} &=& 4\\
& & & & 0 &=& 1\\
& & 2 \, x_{2} &-& 6 \, x_{3} &=& 7
\end{alignat*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-14" id="answer-14"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-14"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\mathrm{RREF} \left[\begin{array}{ccc|c}
1 & -1 & -1 & -6 \\
0 & 1 & -3 & 4 \\
0 & 0 & 0 & 1 \\
0 & 2 & -6 & 7
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & -4 & 0 \\
0 & 1 & -3 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-104">The solution set is \(\left\{\right\} \)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-15"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-105">Show how to find the solution set for the following vector equation</p>
<div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
-3 \\
0 \\
-1 \\
-2
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
7 \\
-5 \\
3 \\
7
\end{array}\right] = \left[\begin{array}{c}
1 \\
6 \\
0 \\
-2
\end{array}\right] .
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-15" id="answer-15"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-15"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\mathrm{RREF} \left[\begin{array}{ccc|c}
-3 & -1 & 7 & 1 \\
0 & 1 & -5 & 6 \\
-1 & 0 & 3 & 0 \\
-2 & -1 & 7 & -2
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & -3 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-106">The solution set is \(\left\{ \left[\begin{array}{c}
-3 \\
1 \\
-1
\end{array}\right] \right\} \)</p>
</div></div>
</div></article><p id="p-107"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/E3/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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