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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="V9"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.9</span> <span class="title">Homogeneous Linear Systems (V9)</span>
</h2>
<article class="definition definition-like" id="definition-16"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.9.1</span><span class="period">.</span>
</h6>
<p id="p-419">A <dfn class="terminology">homogeneous</dfn> system of linear equations is one of the form:</p>
<div class="displaymath">
\begin{alignat*}{5}
a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& 0 \\
a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& 0 \\
\vdots& &\vdots& && &\vdots&&\vdots\\
a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& 0
\end{alignat*}
</div>
<p id="p-420">This system is equivalent to the vector equation:</p>
<div class="displaymath">
\begin{equation*}
x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0}
\end{equation*}
</div>
<p data-braille="continuation">and the augmented matrix:</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & 0\\
a_{21} & a_{22} & \cdots & a_{2n} & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn} & 0
\end{array}\right]
\end{equation*}
</div></article><article class="activity project-like" id="activity-62"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.9.1</span><span class="period">.</span>
</h6>
<p id="p-421">Note that if \(\left[\begin{array}{c} a_1 \\ \vdots \\ a_n \end{array}\right] \) and \(\left[\begin{array}{c} b_1 \\ \vdots \\ b_n \end{array}\right] \) are solutions to \(x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0}\) so is \(\left[\begin{array}{c} a_1 +b_1\\ \vdots \\ a_n+b_n \end{array}\right] \text{,}\) since</p>
<div class="displaymath">
\begin{equation*}
a_1 \vec{v}_1+\cdots+a_n \vec{v}_n = \vec{0}
\text{ and }
b_1 \vec{v}_1+\cdots+b_n \vec{v}_n = \vec{0}
\end{equation*}
</div>
<p data-braille="continuation">implies</p>
<div class="displaymath">
\begin{equation*}
(a_1 + b_1) \vec{v}_1+\cdots+(a_n+b_n) \vec{v}_n = \vec{0} .
\end{equation*}
</div>
<p id="p-422">Similarly, if \(c \in \IR\text{,}\) \(\left[\begin{array}{c} ca_1 \\ \vdots \\ ca_n \end{array}\right] \) is a solution. Thus the solution set of a homogeneous system is...</p>
<ol class="lower-alpha">
<li id="li-302"><p id="p-423">A basis for \(\IR^n\text{.}\)</p></li>
<li id="li-303"><p id="p-424">A subspace of \(\IR^n\text{.}\)</p></li>
<li id="li-304"><p id="p-425">The empty set.</p></li>
</ol></article><article class="activity project-like" id="activity-63"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.9.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-23">
<p id="p-426">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{5}
x_1&\,+\,&2x_2&\,\,& &\,+\,& x_4 &=& 0\\
2x_1&\,+\,&4x_2&\,-\,&x_3 &\,-\,&2 x_4 &=& 0\\
3x_1&\,+\,&6x_2&\,-\,&x_3 &\,-\,& x_4 &=& 0
\end{alignat*}
</div>
</div>
<article class="task exercise-like" id="task-53"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-427">Find its solution set (a subspace of \(\IR^4\)).</p></article><article class="task exercise-like" id="task-54"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-428">Rewrite this solution space in the form</p>
<div class="displaymath">
\begin{equation*}
\setBuilder{ a \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right] + b \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right] }{a,b \in \IR}.
\end{equation*}
</div></article><article class="task exercise-like" id="task-55"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-429">Rewrite this solution space in the form</p>
<div class="displaymath">
\begin{equation*}
\vspan\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}.
\end{equation*}
</div></article></article><article class="fact theorem-like" id="fact-12"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.9.2</span><span class="period">.</span>
</h6>
<p id="p-430">The coefficients of the free variables in the solution set of a linear system always yield linearly independent vectors.</p>
<p id="p-431">Thus if</p>
<div class="displaymath">
\begin{equation*}
\setBuilder{
a \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right] +
b \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right]
}{
a,b \in \IR
} = \vspan\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right],
\left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right] \right\}
\end{equation*}
</div>
<p data-braille="continuation">is the solution space for a homogeneous system, then</p>
<div class="displaymath">
\begin{equation*}
\setList{
\left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right],
\left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right]
}
\end{equation*}
</div>
<p data-braille="continuation">is a basis for the solution space.</p></article><article class="activity project-like" id="activity-64"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.9.3</span><span class="period">.</span>
</h6>
<p id="p-432">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{5}
2x_1&\,+\,&4x_2&\,+\,& 2x_3&\,-\,&4x_4 &=& 0 \\
-2x_1&\,-\,&4x_2&\,+\,&x_3 &\,+\,& x_4 &=& 0\\
3x_1&\,+\,&6x_2&\,-\,&x_3 &\,-\,&4 x_4 &=& 0
\end{alignat*}
</div>
<p id="p-433">Find a basis for its solution space.</p></article><article class="activity project-like" id="activity-65"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.9.4</span><span class="period">.</span>
</h6>
<p id="p-434">Consider the homogeneous vector equation</p>
<div class="displaymath">
\begin{equation*}
x_1 \left[\begin{array}{c} 2 \\ -2 \\ 3 \end{array}\right]+
x_2 \left[\begin{array}{c} 4 \\ -4 \\ 6 \end{array}\right]+
x_3 \left[\begin{array}{c} 2 \\ 1 \\ -1 \end{array}\right]+
x_4 \left[\begin{array}{c} -4 \\ 1 \\ -4 \end{array}\right]=
\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]
\end{equation*}
</div>
<p id="p-435">Find a basis for its solution space.</p></article><article class="activity project-like" id="activity-66"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.9.5</span><span class="period">.</span>
</h6>
<p id="p-436">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{5}
x_1&\,-\,&3x_2&\,+\,& 2x_3 &=& 0\\
2x_1&\,+\,&6x_2&\,+\,&4x_3 &=& 0\\
x_1&\,+\,&6x_2&\,-\,&4x_3 &=& 0
\end{alignat*}
</div>
<p id="p-437">Find a basis for its solution space.</p></article><article class="observation remark-like" id="observation-15"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.9.3</span><span class="period">.</span>
</h6>
<p id="p-438">The basis of the trivial vector space is the empty set. You can denote this as either \(\emptyset\) or \(\{\}\text{.}\)</p>
<p id="p-439">Thus, if \(\vec{0}\) is the only solution of a homogeneous system, the basis of the solution space is \(\emptyset\text{.}\)</p></article><section class="exercises" id="exercises-12"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.9.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-56"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-440">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{6}
x_{1} &-& 2 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &+& 4 \, x_{5} &=& 0\\
2 \, x_{1} &-& 3 \, x_{2} &-& 3 \, x_{3} &+& 5 \, x_{4} &+& 6 \, x_{5} &=& 0\\
&-& 5 \, x_{2} &+& 5 \, x_{3} &+& 5 \, x_{4} &+& 10 \, x_{5} &=& 0
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-305">Find the solution space of this system.</li>
<li id="li-306">Find a basis of the solution space.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-56" id="answer-56"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-56"><div class="answer solution-like">
<div class="displaymath" id="p-441">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccccc|c}
1 & -2 & -1 & 3 & 4 & 0 \\
2 & -3 & -3 & 5 & 6 & 0 \\
0 & -5 & 5 & 5 & 10 & 0
\end{array}\right] = \left[\begin{array}{ccccc|c}
1 & 0 & -3 & 1 & 0 & 0 \\
0 & 1 & -1 & -1 & -2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-307">The solution space is \(\left\{ \left[\begin{array}{c}
3 \, a - b \\
a + b + 2 \, c \\
a \\
b \\
c
\end{array}\right] \middle|\,a\text{\texttt{,}}b\text{\texttt{,}}c\in\mathbb{R}\right\} \)</li>
<li id="li-308">A basis of the solution space is \(\left\{ \left[\begin{array}{c}
3 \\
1 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
-1 \\
1 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
0 \\
2 \\
0 \\
0 \\
1
\end{array}\right] \right\} \text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-57"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-442">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{5}
& & & & x_{3} &-& 2 \, x_{4} &=& 0\\
x_{1} &-& 3 \, x_{2} &+& 5 \, x_{3} &-& 12 \, x_{4} &=& 0\\
& & &-& 4 \, x_{3} &+& 8 \, x_{4} &=& 0\\
x_{1} &-& 3 \, x_{2} &+& 3 \, x_{3} &-& 8 \, x_{4} &=& 0
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-309">Find the solution space of this system.</li>
<li id="li-310">Find a basis of the solution space.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-57" id="answer-57"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-57"><div class="answer solution-like">
<div class="displaymath" id="p-443">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc|c}
0 & 0 & 1 & -2 & 0 \\
1 & -3 & 5 & -12 & 0 \\
0 & 0 & -4 & 8 & 0 \\
1 & -3 & 3 & -8 & 0
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & -3 & 0 & -2 & 0 \\
0 & 0 & 1 & -2 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-311">The solution space is \(\left\{ \left[\begin{array}{c}
3 \, a + 2 \, b \\
a \\
2 \, b \\
b
\end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\} \)</li>
<li id="li-312">A basis of the solution space is \(\left\{ \left[\begin{array}{c}
3 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
2 \\
0 \\
2 \\
1
\end{array}\right] \right\} \text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-58"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-444">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{4}
3 \, x_{1} &-& 10 \, x_{2} & & &=& 0\\
-2 \, x_{1} &+& 7 \, x_{2} & & &=& 0\\
x_{1} &+& 2 \, x_{2} & & &=& 0\\
2 \, x_{1} &-& 7 \, x_{2} & & &=& 0\\
3 \, x_{1} &-& 9 \, x_{2} & & &=& 0
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-313">Find the solution space of this system.</li>
<li id="li-314">Find a basis of the solution space.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-58" id="answer-58"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-58"><div class="answer solution-like">
<div class="displaymath" id="p-445">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc|c}
3 & -10 & 0 & 0 \\
-2 & 7 & 0 & 0 \\
1 & 2 & 0 & 0 \\
2 & -7 & 0 & 0 \\
3 & -9 & 0 & 0
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-315">The solution space is \(\left\{ \left[\begin{array}{c}
0 \\
0 \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \)</li>
<li id="li-316">A basis of the solution space is \(\left\{ \left[\begin{array}{c}
0 \\
0 \\
1
\end{array}\right] \right\} \text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-59"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-446">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{5}
x_{1} &-& x_{2} &-& x_{3} &-& 8 \, x_{4} &=& 0\\
2 \, x_{1} &-& x_{2} &-& 5 \, x_{3} &-& 12 \, x_{4} &=& 0\\
&-& 3 \, x_{2} &+& 9 \, x_{3} &-& 11 \, x_{4} &=& 0\\
-x_{1} &+& x_{2} &+& x_{3} &+& 7 \, x_{4} &=& 0
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-317">Find the solution space of this system.</li>
<li id="li-318">Find a basis of the solution space.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-59" id="answer-59"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-59"><div class="answer solution-like">
<div class="displaymath" id="p-447">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc|c}
1 & -1 & -1 & -8 & 0 \\
2 & -1 & -5 & -12 & 0 \\
0 & -3 & 9 & -11 & 0 \\
-1 & 1 & 1 & 7 & 0
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & -4 & 0 & 0 \\
0 & 1 & -3 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-319">The solution space is \(\left\{ \left[\begin{array}{c}
4 \, a \\
3 \, a \\
a \\
0
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \)</li>
<li id="li-320">A basis of the solution space is \(\left\{ \left[\begin{array}{c}
4 \\
3 \\
1 \\
0
\end{array}\right] \right\} \text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-60"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-448">Consider the homogeneous system of equations</p>
<div class="displaymath">
\begin{alignat*}{4}
2 \, x_{1} &-& 2 \, x_{2} & & &=& 0\\
-5 \, x_{1} &+& x_{2} &+& 4 \, x_{3} &=& 0\\
-2 \, x_{1} &+& 3 \, x_{2} &-& x_{3} &=& 0\\
-5 \, x_{1} &-& 2 \, x_{2} &+& 7 \, x_{3} &=& 0\\
x_{1} &-& 2 \, x_{2} &+& x_{3} &=& 0
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-321">Find the solution space of this system.</li>
<li id="li-322">Find a basis of the solution space.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-60" id="answer-60"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-60"><div class="answer solution-like">
<div class="displaymath" id="p-449">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc|c}
2 & -2 & 0 & 0 \\
-5 & 1 & 4 & 0 \\
-2 & 3 & -1 & 0 \\
-5 & -2 & 7 & 0 \\
1 & -2 & 1 & 0
\end{array}\right] = \left[\begin{array}{ccc|c}
1 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-323">The solution space is \(\left\{ \left[\begin{array}{c}
a \\
a \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} \)</li>
<li id="li-324">A basis of the solution space is \(\left\{ \left[\begin{array}{c}
1 \\
1 \\
1
\end{array}\right] \right\} \text{.}\)</li>
</ol>
</div></div>
</div></article><p id="p-450"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V9/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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