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<!DOCTYPE html>
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\newcommand{\IR}{\mathbb{R}}
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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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<li class="link frontmatter"><a href="frontmatter.html" data-scroll="frontmatter"><span class="title">Front Matter</span></a></li>
<li class="link">
<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>
<li><a href="E3.html" data-scroll="E3">Solving Linear Systems (E3)</a></li>
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<li class="link">
<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" class="active">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>
<li><a href="V7.html" data-scroll="V7">Subspace Basis and Dimension (V7)</a></li>
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<li><a href="V9.html" data-scroll="V9">Homogeneous Linear Systems (V9)</a></li>
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<a href="A.html" data-scroll="A"><span class="codenumber">3</span> <span class="title">Algebraic Properties of Linear Maps (A)</span></a><ul>
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<li><a href="A2.html" data-scroll="A2">Standard Matrices (A2)</a></li>
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<a href="G.html" data-scroll="G"><span class="codenumber">5</span> <span class="title">Geometric Properties of Linear Maps (G)</span></a><ul>
<li><a href="G1.html" data-scroll="G1">Row Operations and Determinants (G1)</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="V2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.2</span> <span class="title">Linear Combinations (V2)</span>
</h2>
<article class="definition definition-like" id="definition-10"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.2.1</span><span class="period">.</span>
</h6>
<p id="p-175">A <dfn class="terminology">linear combination</dfn> of a set of vectors \(\{\vec v_1,\vec v_2,\dots,\vec v_m\}\) is given by \(c_1\vec v_1+c_2\vec v_2+\dots+c_m\vec v_m\) for any choice of scalar multiples \(c_1,c_2,\dots,c_m\text{.}\)</p>
<p id="p-176">For example, we can say \(\left[\begin{array}{c}3 \\0 \\ 5\end{array}\right]\) is a linear combination of the vectors \(\left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right]\) and \(\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]\) since</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{c} 3 \\ 0 \\ 5 \end{array}\right] =
2 \left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right] +
1\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]\text{.}
\end{equation*}
</div></article><article class="definition definition-like" id="definition-11"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.2.2</span><span class="period">.</span>
</h6>
<p id="p-177">The <dfn class="terminology">span</dfn> of a set of vectors is the collection of all linear combinations of that set:</p>
<div class="displaymath">
\begin{equation*}
\vspan\{\vec v_1,\vec v_2,\dots,\vec v_m\} =
\setBuilder{c_1\vec v_1+c_2\vec v_2+\dots+c_m\vec v_m}{
c_i\in\IR}\text{.}
\end{equation*}
</div>
<p id="p-178">For example:</p>
<div class="displaymath">
\begin{equation*}
\vspan\setList
{
\left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right],
\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]
} = \setBuilder
{
a\left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right]+
b\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]
}{
a,b\in\IR
}
\end{equation*}
</div></article><article class="activity project-like" id="activity-24"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.1</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-10"><p id="p-179">Consider \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right]\right\}\text{.}\)</p></div>
<article class="task exercise-like" id="task-22"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-180">Sketch \(1\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}1\\2\end{array}\right]\text{,}\) \(3\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}3\\6\end{array}\right]\text{,}\) \(0\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}0\\0\end{array}\right]\text{,}\) and \(-2\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}-2\\-4\end{array}\right]\) in the \(xy\) plane.</p></article><article class="task exercise-like" id="task-23"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-181">Sketch a representation of all the vectors belonging to \(\vspan\setList{\left[\begin{array}{c}1\\2\end{array}\right]}
=
\setBuilder{a\left[\begin{array}{c}1\\2\end{array}\right]}{a\in\IR}\) in the \(xy\) plane.</p></article></article><article class="activity project-like" id="activity-25"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-11"><p id="p-182">Consider \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right],
\left[\begin{array}{c}-1\\1\end{array}\right]\right\}\text{.}\)</p></div>
<article class="task exercise-like" id="task-24"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-183">Sketch the following linear combinations in the \(xy\) plane.</p>
<div class="displaymath">
\begin{equation*}
1\left[\begin{array}{c}1\\2\end{array}\right]+
0\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em}
0\left[\begin{array}{c}1\\2\end{array}\right]+
1\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em}
1\left[\begin{array}{c}1\\2\end{array}\right]+
1\left[\begin{array}{c}-1\\1\end{array}\right]
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
-2\left[\begin{array}{c}1\\2\end{array}\right]+
1\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em}
-1\left[\begin{array}{c}1\\2\end{array}\right]+
-2\left[\begin{array}{c}-1\\1\end{array}\right]
\end{equation*}
</div></article><article class="task exercise-like" id="task-25"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-184">Sketch a representation of all the vectors belonging to \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right],
\left[\begin{array}{c}-1\\1\end{array}\right]\right\}\) in the \(xy\) plane.</p></article></article><article class="activity project-like" id="activity-26"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.3</span><span class="period">.</span>
</h6>
<p id="p-185">Sketch a representation of all the vectors belonging to \(\vspan\left\{\left[\begin{array}{c}6\\-4\end{array}\right],
\left[\begin{array}{c}-3\\2\end{array}\right]\right\}\) in the \(xy\) plane.</p></article><article class="activity project-like" id="activity-27"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.4</span><span class="period">.</span>
</h6>
<article class="task exercise-like" id="task-26"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-187">Reinterpret this vector equation as a system of linear equations.</p></article><article class="task exercise-like" id="task-27"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-188">Find its solution set, using technology to find \(\RREF\) of its corresponding augmented matrix.</p></article><article class="task exercise-like" id="task-28"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-189">Given this solution set, does \(\left[\begin{array}{c}-1\\-6\\1\end{array}\right]\) belong to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right],
\left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\text{?}\)</p></article></article><article class="fact theorem-like" id="fact-2"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.2.3</span><span class="period">.</span>
</h6>
<p id="p-190">A vector \(\vec b\) belongs to \(\vspan\{\vec v_1,\dots,\vec v_n\}\) if and only if the vector equation \(x_1 \vec{v}_1+\cdots+x_n \vec{v_n}=\vec{b}\) is consistent.</p></article><article class="observation remark-like" id="observation-6"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.2.4</span><span class="period">.</span>
</h6>
<p id="p-191">The following are all equivalent statements:</p>
<ul class="disc">
<li id="li-126"><p id="p-192">The vector \(\vec{b}\) belongs to \(\vspan\{\vec v_1,\dots,\vec v_n\}\text{.}\)</p></li>
<li id="li-127"><p id="p-193">The vector equation \(x_1 \vec{v}_1+\cdots+x_n \vec{v}_n=\vec{b}\) is consistent.</p></li>
<li id="li-128"><p id="p-194">The linear system corresponding to \(\left[\vec v_1\,\dots\,\vec v_n \,|\, \vec b\right]\) is consistent.</p></li>
<li id="li-129"><p id="p-195">\(\RREF\left[\vec v_1\,\dots\,\vec v_n \,|\, \vec b\right]\) doesn't have a row \([0\,\cdots\,0\,|\,1]\) representing the contradiction \(0=1\text{.}\)</p></li>
</ul></article><article class="activity project-like" id="activity-28"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.5</span><span class="period">.</span>
</h6>
<p id="p-196">Determine if \(\left[\begin{array}{c}3\\-2\\1 \\ 5\end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3 \\ 2\end{array}\right],
\left[\begin{array}{c}-1\\-3\\2 \\ 2\end{array}\right]\right\}\) by solving an appropriate vector equation.</p></article><article class="activity project-like" id="activity-29"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.6</span><span class="period">.</span>
</h6>
<p id="p-197">Determine if \(\left[\begin{array}{c}-1\\-9\\0\end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right],
\left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\) by solving an appropriate vector equation.</p></article><article class="activity project-like" id="activity-30"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.7</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-12"><p id="p-198">Does the third-degree polynomial \(3y^3-2y^2+y+5\) in \(\P^3\) belong to \(\vspan\{y^3-3y+2,-y^3-3y^2+2y+2\}\text{?}\)</p></div>
<article class="task exercise-like" id="task-29"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-199">Reinterpret this question as a question about the solution(s) of a polynomial equation.</p></article><article class="task exercise-like" id="task-30"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-200">Answer this equivalent question, and use its solution to answer the original question.</p></article></article><article class="activity project-like" id="activity-31"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.8</span><span class="period">.</span>
</h6>
<p id="p-201">Does the polynomial \(x^2+x+1\) belong to \(\vspan\{x^2-x,x+1, x^2-1\}\text{?}\)</p></article><article class="activity project-like" id="activity-32"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.2.9</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-13"><p id="p-202">Does the matrix \(\left[\begin{array}{c}3&-2\\1&5\end{array}\right]\) belong to \(\vspan\left\{\left[\begin{array}{c}1&0\\-3&2\end{array}\right],
\left[\begin{array}{c}-1&-3\\2&2\end{array}\right]\right\}\text{?}\)</p></div>
<article class="task exercise-like" id="task-31"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-203">Reinterpret this question as a question about the solution(s) of a matrix equation.</p></article><article class="task exercise-like" id="task-32"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-204">Answer this equivalent question, and use its solution to answer the original question.</p></article></article><section class="exercises" id="exercises-5"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.2.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-21"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-205">Consider the statement <article class="claim theorem-like" id="claim-1"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.5</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-2 \\
-6 \\
-6 \\
-4
\end{array}\right] \)\(\left[\begin{array}{c}
3 \\
2 \\
-5 \\
-3
\end{array}\right] , \left[\begin{array}{c}
0 \\
1 \\
-3 \\
1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
4 \\
-3 \\
-3 \\
0
\end{array}\right] \text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-130">Write an equivalent statement using a vector equation.</li>
<li id="li-131">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-21" id="answer-21"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-21"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc|c}
3 & 0 & 4 & -2 \\
2 & 1 & -3 & -6 \\
-5 & -3 & -3 & -6 \\
-3 & 1 & 0 & -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>
<ol class="lower-alpha">
<li id="li-132">The statement <article class="claim theorem-like" id="claim-2"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.6</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-2 \\
-6 \\
-6 \\
-4
\end{array}\right] \)\(\left[\begin{array}{c}
3 \\
2 \\
-5 \\
-3
\end{array}\right] , \left[\begin{array}{c}
0 \\
1 \\
-3 \\
1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
4 \\
-3 \\
-3 \\
0
\end{array}\right] \text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-3"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.7</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
3 \\
2 \\
-5 \\
-3
\end{array}\right] + x_{2} \left[\begin{array}{c}
0 \\
1 \\
-3 \\
1
\end{array}\right] + x_{3} \left[\begin{array}{c}
4 \\
-3 \\
-3 \\
0
\end{array}\right] = \left[\begin{array}{c}
-2 \\
-6 \\
-6 \\
-4
\end{array}\right] \)</article>
</li>
<li id="li-133"><p id="p-206">\(\left[\begin{array}{c}
-2 \\
-6 \\
-6 \\
-4
\end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c}
3 \\
2 \\
-5 \\
-3
\end{array}\right] , \left[\begin{array}{c}
0 \\
1 \\
-3 \\
1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
4 \\
-3 \\
-3 \\
0
\end{array}\right] \text{.}\)</p></li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-22"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-207">Consider the statement <article class="claim theorem-like" id="claim-4"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.8</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-6 \\
1 \\
7 \\
-2
\end{array}\right] \)\(\left[\begin{array}{c}
4 \\
-2 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
0 \\
4 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-3 \\
1 \\
-4 \\
-1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-2 \\
-2 \\
2 \\
-2
\end{array}\right] \text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-134">Write an equivalent statement using a vector equation.</li>
<li id="li-135">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-22" id="answer-22"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-22"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc|c}
4 & -5 & -3 & -2 & -6 \\
-2 & 0 & 1 & -2 & 1 \\
1 & 4 & -4 & 2 & 7 \\
0 & -3 & -1 & -2 & -2
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-136">The statement <article class="claim theorem-like" id="claim-5"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.9</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-6 \\
1 \\
7 \\
-2
\end{array}\right] \)\(\left[\begin{array}{c}
4 \\
-2 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
0 \\
4 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-3 \\
1 \\
-4 \\
-1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-2 \\
-2 \\
2 \\
-2
\end{array}\right] \text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-6"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.10</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
4 \\
-2 \\
1 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-5 \\
0 \\
4 \\
-3
\end{array}\right] + x_{3} \left[\begin{array}{c}
-3 \\
1 \\
-4 \\
-1
\end{array}\right] + x_{4} \left[\begin{array}{c}
-2 \\
-2 \\
2 \\
-2
\end{array}\right] = \left[\begin{array}{c}
-6 \\
1 \\
7 \\
-2
\end{array}\right] \)</article>
</li>
<li id="li-137"><p id="p-208">\(\left[\begin{array}{c}
-6 \\
1 \\
7 \\
-2
\end{array}\right] \) is a linear combination of the vectors \(\left[\begin{array}{c}
4 \\
-2 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
0 \\
4 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-3 \\
1 \\
-4 \\
-1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-2 \\
-2 \\
2 \\
-2
\end{array}\right] \text{.}\)</p></li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-23"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-209">Consider the statement <article class="claim theorem-like" id="claim-7"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.11</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-7 \\
6 \\
-1 \\
-9
\end{array}\right] \)\(\left[\begin{array}{c}
2 \\
-3 \\
-3 \\
0
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
2
\end{array}\right] , \left[\begin{array}{c}
8 \\
-2 \\
-6 \\
4
\end{array}\right] , \left[\begin{array}{c}
12 \\
2 \\
-6 \\
8
\end{array}\right] , \text{ and } \left[\begin{array}{c}
0 \\
-5 \\
-3 \\
-2
\end{array}\right] \text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-138">Write an equivalent statement using a vector equation.</li>
<li id="li-139">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-23" id="answer-23"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-23"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccccc|c}
2 & 4 & 8 & 12 & 0 & -7 \\
-3 & -1 & -2 & 2 & -5 & 6 \\
-3 & -3 & -6 & -6 & -3 & -1 \\
0 & 2 & 4 & 8 & -2 & -9
\end{array}\right] = \left[\begin{array}{ccccc|c}
1 & 0 & 0 & -2 & 2 & 0 \\
0 & 1 & 2 & 4 & -1 & 0 \\
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-140">The statement <article class="claim theorem-like" id="claim-8"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.12</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-7 \\
6 \\
-1 \\
-9
\end{array}\right] \)\(\left[\begin{array}{c}
2 \\
-3 \\
-3 \\
0
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
2
\end{array}\right] , \left[\begin{array}{c}
8 \\
-2 \\
-6 \\
4
\end{array}\right] , \left[\begin{array}{c}
12 \\
2 \\
-6 \\
8
\end{array}\right] , \text{ and } \left[\begin{array}{c}
0 \\
-5 \\
-3 \\
-2
\end{array}\right] \text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-9"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.13</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
2 \\
-3 \\
-3 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
2
\end{array}\right] + x_{3} \left[\begin{array}{c}
8 \\
-2 \\
-6 \\
4
\end{array}\right] + x_{4} \left[\begin{array}{c}
12 \\
2 \\
-6 \\
8
\end{array}\right] + x_{5} \left[\begin{array}{c}
0 \\
-5 \\
-3 \\
-2
\end{array}\right] = \left[\begin{array}{c}
-7 \\
6 \\
-1 \\
-9
\end{array}\right] \)</article>
</li>
<li id="li-141"><p id="p-210">\(\left[\begin{array}{c}
-7 \\
6 \\
-1 \\
-9
\end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c}
2 \\
-3 \\
-3 \\
0
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
2
\end{array}\right] , \left[\begin{array}{c}
8 \\
-2 \\
-6 \\
4
\end{array}\right] , \left[\begin{array}{c}
12 \\
2 \\
-6 \\
8
\end{array}\right] , \text{ and } \left[\begin{array}{c}
0 \\
-5 \\
-3 \\
-2
\end{array}\right] \text{.}\)</p></li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-24"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-211">Consider the statement <article class="claim theorem-like" id="claim-10"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.14</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \)\(\left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] , \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] \text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-142">Write an equivalent statement using a vector equation.</li>
<li id="li-143">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-24" id="answer-24"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-24"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc|c}
2 & -4 & -1 & -1 & 7 \\
2 & -2 & -2 & 2 & 10 \\
2 & -4 & -4 & -3 & 16 \\
-1 & 1 & -3 & -2 & 7
\end{array}\right] = \left[\begin{array}{cccc|c}
1 & 0 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & -3 \\
0 & 0 & 0 & 1 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-144">The statement <article class="claim theorem-like" id="claim-11"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.15</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \)\(\left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] , \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] \text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-12"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.16</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] + x_{2} \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] + x_{4} \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] = \left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \)</article>
</li>
<li id="li-145"><p id="p-212">\(\left[\begin{array}{c}
7 \\
10 \\
16 \\
7
\end{array}\right] \) is a linear combination of the vectors \(\left[\begin{array}{c}
2 \\
2 \\
2 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-2 \\
-4 \\
1
\end{array}\right] , \left[\begin{array}{c}
-1 \\
-2 \\
-4 \\
-3
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-1 \\
2 \\
-3 \\
-2
\end{array}\right] \text{.}\)</p></li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-25"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-213">Consider the statement <article class="claim theorem-like" id="claim-13"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.17</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-5 \\
-4 \\
2 \\
3
\end{array}\right] \)\(\left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-2 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-2 \\
-6 \\
2
\end{array}\right] , \left[\begin{array}{c}
-4 \\
8 \\
22 \\
-2
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-4 \\
2 \\
5 \\
1
\end{array}\right] \text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-146">Write an equivalent statement using a vector equation.</li>
<li id="li-147">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-25" id="answer-25"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-25"><div class="answer solution-like">
<div class="displaymath">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccccc|c}
-1 & 4 & -2 & -4 & -4 & -5 \\
-1 & -2 & -2 & 8 & 2 & -4 \\
-3 & -5 & -6 & 22 & 5 & 2 \\
1 & -1 & 2 & -2 & 1 & 3
\end{array}\right] = \left[\begin{array}{ccccc|c}
1 & 0 & 2 & -4 & 0 & 0 \\
0 & 1 & 0 & -2 & -1 & 0 \\
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-148">The statement <article class="claim theorem-like" id="claim-14"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.18</span><span class="period">.</span>
</h6>\(\left[\begin{array}{c}
-5 \\
-4 \\
2 \\
3
\end{array}\right] \)\(\left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-2 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-2 \\
-6 \\
2
\end{array}\right] , \left[\begin{array}{c}
-4 \\
8 \\
22 \\
-2
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-4 \\
2 \\
5 \\
1
\end{array}\right] \text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-15"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.2.19</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
1
\end{array}\right] + x_{2} \left[\begin{array}{c}
4 \\
-2 \\
-5 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
-2 \\
-2 \\
-6 \\
2
\end{array}\right] + x_{4} \left[\begin{array}{c}
-4 \\
8 \\
22 \\
-2
\end{array}\right] + x_{5} \left[\begin{array}{c}
-4 \\
2 \\
5 \\
1
\end{array}\right] = \left[\begin{array}{c}
-5 \\
-4 \\
2 \\
3
\end{array}\right] \)</article>
</li>
<li id="li-149"><p id="p-214">\(\left[\begin{array}{c}
-5 \\
-4 \\
2 \\
3
\end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-2 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-2 \\
-6 \\
2
\end{array}\right] , \left[\begin{array}{c}
-4 \\
8 \\
22 \\
-2
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-4 \\
2 \\
5 \\
1
\end{array}\right] \text{.}\)</p></li>
</ol>
</div></div>
</div></article><p id="p-215"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V2/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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