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<!DOCTYPE html>
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<!--* on 2021-01-14T10:40:32-06:00 *-->
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<html lang="en-US">
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<title>Spanning Sets (V3)</title>
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<a class="assistive" href="#content">Skip to main content</a><div xmlns:svg="http://www.w3.org/2000/svg" class="hidden-content" style="display:none">\(\require{enclose}
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IC}{\mathbb{C}}
\renewcommand{\P}{\mathcal{P}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\RREF}{\operatorname{RREF}}
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\)</div>
<header id="masthead" class="smallbuttons"><div class="banner"><div class="container">
<a id="logo-link" href=""></a><div class="title-container">
<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>
</div>
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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>
</ul>
</li>
<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">Linear Combinations (V2)</a></li>
<li><a href="V3.html" data-scroll="V3" class="active">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>
<li><a href="V8.html" data-scroll="V8">Polynomial and Matrix Spaces (V8)</a></li>
<li><a href="V9.html" data-scroll="V9">Homogeneous Linear Systems (V9)</a></li>
</ul>
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<li class="link">
<a href="A.html" data-scroll="A"><span class="codenumber">3</span> <span class="title">Algebraic Properties of Linear Maps (A)</span></a><ul>
<li><a href="A1.html" data-scroll="A1">Linear Transformations (A1)</a></li>
<li><a href="A2.html" data-scroll="A2">Standard Matrices (A2)</a></li>
<li><a href="A3.html" data-scroll="A3">Image and Kernel (A3)</a></li>
<li><a href="A4.html" data-scroll="A4">Injective and Surjective Linear Maps (A4)</a></li>
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<a href="M.html" data-scroll="M"><span class="codenumber">4</span> <span class="title">Matrices (M)</span></a><ul>
<li><a href="M1.html" data-scroll="M1">Matrices and Multiplication (M1)</a></li>
<li><a href="M2.html" data-scroll="M2">Row Operations as Matrix Multiplication (M2)</a></li>
<li><a href="M3.html" data-scroll="M3">The Inverse of a Matrix (M3)</a></li>
<li><a href="M4.html" data-scroll="M4">Invertible Matrices (M4)</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>
<li><a href="G2.html" data-scroll="G2">Computing Determinants (G2)</a></li>
<li><a href="G3.html" data-scroll="G3">Eigenvalues and Characteristic Polynomials (G3)</a></li>
<li><a href="G4.html" data-scroll="G4">Eigenvectors and Eigenspaces (G4)</a></li>
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</ul></nav><div class="extras"><nav><a class="mathbook-link" href="https://pretextbook.org">Authored in PreTeXt</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://www.mathjax.org/badge/badge.gif" alt="Powered by MathJax"></a></nav></div>
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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="V3"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.3</span> <span class="title">Spanning Sets (V3)</span>
</h2>
<article class="observation remark-like" id="observation-7"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">2.3.1</span><span class="period">.</span>
</h6>
<p id="p-216">Any single non-zero vector/number \(x\) in \(\IR^1\) spans \(\IR^1\text{,}\) since \(\IR^1=\setBuilder{cx}{c\in\IR}\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}
\draw[<->] (-3,0) -- (3,0);
\draw[thick,->,blue] (0,0) -- (2,0) node[above] {x};
\draw (0,-0.2) -- (0,0.2) node[above] {0};
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-33"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.1</span><span class="period">.</span>
</h6>
<p id="p-217">How many vectors are required to span \(\IR^2\text{?}\) Sketch a drawing in the \(xy\) plane to support your answer.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\draw[<->] (-4,0) -- (4,0);
\draw[<->] (0,-4) -- (0,4);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-150"><p id="p-218">\(\displaystyle 1\)</p></li>
<li id="li-151"><p id="p-219">\(\displaystyle 2\)</p></li>
<li id="li-152"><p id="p-220">\(\displaystyle 3\)</p></li>
<li id="li-153"><p id="p-221">\(\displaystyle 4\)</p></li>
<li id="li-154"><p id="p-222">Infinitely Many</p></li>
</ol></article><article class="activity project-like" id="activity-34"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.2</span><span class="period">.</span>
</h6>
<p id="p-223">How many vectors are required to span \(\IR^3\text{?}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x={(210:0.8cm)}, y={(0:1cm)}, z={(90:1cm)},scale=0.4]
\draw[->] (0,0,0) -- (6,0,0);
\draw[->] (0,0,0) -- (0,6,0);
\draw[->] (0,0,0) -- (0,0,6);
\end{tikzpicture}
</pre>
<ol class="lower-alpha">
<li id="li-155"><p id="p-224">\(\displaystyle 1\)</p></li>
<li id="li-156"><p id="p-225">\(\displaystyle 2\)</p></li>
<li id="li-157"><p id="p-226">\(\displaystyle 3\)</p></li>
<li id="li-158"><p id="p-227">\(\displaystyle 4\)</p></li>
<li id="li-159"><p id="p-228">Infinitely Many</p></li>
</ol></article><article class="fact theorem-like" id="fact-3"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.3.2</span><span class="period">.</span>
</h6>
<p id="p-229">At least \(n\) vectors are required to span \(\IR^n\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.5]
\draw[<->] (-4,0) -- (4,0);
\draw[<->] (0,-4) -- (0,4);
\draw[blue!50] (2,-4) -- (-2,4);
\draw[thick,blue,->] (0,0) -- (1,-2);
\end{tikzpicture}
\begin{tikzpicture}[x={(210:0.8cm)}, y={(0:1cm)}, z={(90:1cm)},scale=0.4]
\draw[->] (0,0,0) -- (6,0,0);
\draw[->] (0,0,0) -- (0,6,0);
\draw[->] (0,0,0) -- (0,0,6);
\draw[fill=purple!20,fill opacity=0.5]
(-2,-2,2) -- (6,-2,-2) -- (2,2,-2) -- (-6,2,2) -- (-2,-2,2);
\draw[thick,blue,->] (0,0,0) -- (1,-1,0);
\draw[thick,red,->] (0,0,0) -- (-2,0,1);
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-35"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.3</span><span class="period">.</span>
</h6>
<p id="p-230">Choose any vector \(\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]\) in \(\IR^3\) that is not in \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\) by using technology to verify that \(\RREF
\left[\begin{array}{cc|c}1&-2&\unknown\\-1&0&\unknown\\0&1&\unknown\end{array}\right]
=
\left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right]
\text{.}\) (Why does this work?)</p></article><article class="fact theorem-like" id="fact-4"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.3.3</span><span class="period">.</span>
</h6>
<p id="p-231">The set \(\{\vec v_1,\dots,\vec v_m\}\) fails to span all of \(\IR^n\) exactly when the vector equation</p>
<div class="displaymath">
\begin{equation*}
x_1 \vec{v}_1 + \cdots x_m\vec{v}_m = \vec{w}
\end{equation*}
</div>
<p data-braille="continuation">is inconsistent for <em class="emphasis">some</em> vector \(\vec{w}\text{.}\)</p>
<p id="p-232">Note that this happens exactly when \(\RREF[\vec v_1\,\dots\,\vec v_m]\) has a non-pivot row of zeros.</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc}1&-2\\-1&0\\0&1\end{array}\right]\sim
\left[\begin{array}{cc}1&0\\0&1\\0&0\end{array}\right]
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
\Rightarrow
\left[\begin{array}{cc|c}1&-2&a\\-1&0&b\\0&1&c\end{array}\right]\sim
\left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right]
\text{for some choice of vector} \left[\begin{array}{c} a \\ b \\ c \end{array}\right] \text{.}
\end{equation*}
</div></article><article class="activity project-like" id="activity-36"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.4</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-14"><p id="p-233">Consider the set of vectors \(S=\left\{
\left[\begin{array}{c}2\\3\\0\\-1\end{array}\right],
\left[\begin{array}{c}1\\-4\\3\\0\end{array}\right],
\left[\begin{array}{c}1\\7\\-3\\-1\end{array}\right],
\left[\begin{array}{c}0\\3\\5\\7\end{array}\right],
\left[\begin{array}{c}3\\13\\7\\16\end{array}\right]
\right\}\) and the question “Does \(\IR^4=\vspan S\text{?}\)”</p></div>
<article class="task exercise-like" id="task-33"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-234">Rewrite this question in terms of the solutions to a vector equation.</p></article><article class="task exercise-like" id="task-34"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-235">Answer your new question, and use this to answer the original question.</p></article></article><article class="activity project-like" id="activity-37"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.5</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-15">
<p id="p-236">Consider the set of third-degree polynomials</p>
<div class="displaymath">
\begin{align*}
S=\{
&2x^3+3x^2-1,
2x^3+3,
3x^3+13x^2+7x+16,\\
&-x^3+10x^2+7x+14,
4x^3+3x^2+2 \} .
\end{align*}
</div>
<p data-braille="continuation">and the question “Does \(\P^3=\vspan S\text{?}\)”</p>
</div>
<article class="task exercise-like" id="task-35"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-237">Rewrite this question to be about the solutions to a polynomial equation.</p></article><article class="task exercise-like" id="task-36"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-238">Answer your new question, and use this to answer the original question.</p></article></article><article class="activity project-like" id="activity-38"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.6</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-16">
<p id="p-239">Consider the set of matrices</p>
<div class="displaymath">
\begin{equation*}
S = \left\{
\left[\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}\right],
\left[\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array}\right],
\left[\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array}\right]
\right\}
\end{equation*}
</div>
<p data-braille="continuation">and the question “Does \(M_{2,2} = \vspan S\text{?}\)”</p>
</div>
<article class="task exercise-like" id="task-37"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-240">Rewrite this as a question about the solutions to a matrix equation.</p></article><article class="task exercise-like" id="task-38"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-241">Answer your new question, and use this to answer the original question.</p></article></article><article class="activity project-like" id="activity-39"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.3.7</span><span class="period">.</span>
</h6>
<p id="p-242">Let \(\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \IR^7\) be three vectors, and suppose \(\vec{w}\) is another vector with \(\vec{w} \in \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}\text{.}\) What can you conclude about \(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{?}\)</p>
<ol class="lower-alpha">
<li id="li-160">\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is larger than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)</li>
<li id="li-161">\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} = \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)</li>
<li id="li-162">\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is smaller than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)</li>
</ol></article><section class="exercises" id="exercises-6"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.3.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-26"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-243">Consider the statement <article class="claim theorem-like" id="claim-16"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.4</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
3 \\
-5 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-3 \\
3 \\
4
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-3 \\
2 \\
4
\end{array}\right] , \left[\begin{array}{c}
-10 \\
7 \\
12 \\
4
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-163">Write an equivalent statement using a vector equation.</li>
<li id="li-164">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-26" id="answer-26"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-26"><div class="answer solution-like">
<div class="displaymath" id="p-244">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
3 & 3 & -4 & -10 \\
-5 & -3 & -3 & 7 \\
-5 & 3 & 2 & 12 \\
0 & 4 & 4 & 4
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & -2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-165">The statement <article class="claim theorem-like" id="claim-17"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.5</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
3 \\
-5 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-3 \\
3 \\
4
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-3 \\
2 \\
4
\end{array}\right] , \left[\begin{array}{c}
-10 \\
7 \\
12 \\
4
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-18"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.6</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
3 \\
-5 \\
-5 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
3 \\
-3 \\
3 \\
4
\end{array}\right] + x_{3} \left[\begin{array}{c}
-4 \\
-3 \\
2 \\
4
\end{array}\right] + x_{4} \left[\begin{array}{c}
-10 \\
7 \\
12 \\
4
\end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\)</article>
</li>
<li id="li-166">The set of vectors \(\left\{ \left[\begin{array}{c}
3 \\
-5 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-3 \\
3 \\
4
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-3 \\
2 \\
4
\end{array}\right] , \left[\begin{array}{c}
-10 \\
7 \\
12 \\
4
\end{array}\right] \right\} \) does not span \(\mathbb{R}^4\text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-27"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-245">Consider the statement <article class="claim theorem-like" id="claim-19"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.7</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
4 \\
-3 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
1 \\
4 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-3 \\
4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-3 \\
2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
1 \\
-1 \\
3 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-167">Write an equivalent statement using a vector equation.</li>
<li id="li-168">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-27" id="answer-27"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-27"><div class="answer solution-like">
<div class="displaymath" id="p-246">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccccc}
4 & -5 & -3 & -2 & 1 \\
-3 & 1 & 4 & -3 & -1 \\
1 & 4 & -4 & 2 & 3 \\
2 & -4 & -1 & -3 & 0
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & 0 & 0 & \frac{8}{5} \\
0 & 1 & 0 & 0 & \frac{29}{30} \\
0 & 0 & 1 & 0 & \frac{13}{30} \\
0 & 0 & 0 & 1 & -\frac{11}{30}
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-169">The statement <article class="claim theorem-like" id="claim-20"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.8</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
4 \\
-3 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
1 \\
4 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-3 \\
4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-3 \\
2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
1 \\
-1 \\
3 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-21"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.9</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
4 \\
-3 \\
1 \\
2
\end{array}\right] + x_{2} \left[\begin{array}{c}
-5 \\
1 \\
4 \\
-4
\end{array}\right] + x_{3} \left[\begin{array}{c}
-3 \\
4 \\
-4 \\
-1
\end{array}\right] + x_{4} \left[\begin{array}{c}
-2 \\
-3 \\
2 \\
-3
\end{array}\right] + x_{5} \left[\begin{array}{c}
1 \\
-1 \\
3 \\
0
\end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\)</article>
</li>
<li id="li-170">The set of vectors \(\left\{ \left[\begin{array}{c}
4 \\
-3 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
1 \\
4 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-3 \\
4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-3 \\
2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
1 \\
-1 \\
3 \\
0
\end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-28"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-247">Consider the statement <article class="claim theorem-like" id="claim-22"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.10</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
2 \\
-4 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
-1
\end{array}\right] , \left[\begin{array}{c}
0 \\
-2 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
3 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-5 \\
-2 \\
-4
\end{array}\right] , \left[\begin{array}{c}
3 \\
-1 \\
-5 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-171">Write an equivalent statement using a vector equation.</li>
<li id="li-172">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-28" id="answer-28"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-28"><div class="answer solution-like">
<div class="displaymath" id="p-248">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccccc}
2 & 4 & 0 & -5 & -3 & 3 \\
-4 & -1 & -2 & 3 & -5 & -1 \\
-3 & -3 & 1 & -5 & -2 & -5 \\
1 & -1 & 2 & -1 & -4 & 0
\end{array}\right] = \left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & \frac{633}{113} & -\frac{43}{113} \\
0 & 1 & 0 & 0 & -\frac{600}{113} & \frac{180}{113} \\
0 & 0 & 1 & 0 & -\frac{922}{113} & \frac{141}{113} \\
0 & 0 & 0 & 1 & -\frac{159}{113} & \frac{59}{113}
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-173">The statement <article class="claim theorem-like" id="claim-23"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.11</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
2 \\
-4 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
-1
\end{array}\right] , \left[\begin{array}{c}
0 \\
-2 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
3 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-5 \\
-2 \\
-4
\end{array}\right] , \left[\begin{array}{c}
3 \\
-1 \\
-5 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-24"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.12</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
2 \\
-4 \\
-3 \\
1
\end{array}\right] + x_{2} \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
0 \\
-2 \\
1 \\
2
\end{array}\right] + x_{4} \left[\begin{array}{c}
-5 \\
3 \\
-5 \\
-1
\end{array}\right] + x_{5} \left[\begin{array}{c}
-3 \\
-5 \\
-2 \\
-4
\end{array}\right] + x_{6} \left[\begin{array}{c}
3 \\
-1 \\
-5 \\
0
\end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\)</article>
</li>
<li id="li-174">The set of vectors \(\left\{ \left[\begin{array}{c}
2 \\
-4 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
-1
\end{array}\right] , \left[\begin{array}{c}
0 \\
-2 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
3 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-5 \\
-2 \\
-4
\end{array}\right] , \left[\begin{array}{c}
3 \\
-1 \\
-5 \\
0
\end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-29"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-249">Consider the statement <article class="claim theorem-like" id="claim-25"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.13</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
2 \\
2 \\
0 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-4 \\
4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
8 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
4 \\
0 \\
-3
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-175">Write an equivalent statement using a vector equation.</li>
<li id="li-176">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-29" id="answer-29"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-29"><div class="answer solution-like">
<div class="displaymath" id="p-250">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccccc}
2 & -2 & -3 & 1 & -5 \\
2 & -4 & -4 & 0 & 4 \\
0 & 4 & -4 & 8 & 0 \\
-4 & -1 & -1 & 0 & -3
\end{array}\right] = \left[\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-177">The statement <article class="claim theorem-like" id="claim-26"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.14</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
2 \\
2 \\
0 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-4 \\
4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
8 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
4 \\
0 \\
-3
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-27"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.15</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
2 \\
2 \\
0 \\
-4
\end{array}\right] + x_{2} \left[\begin{array}{c}
-2 \\
-4 \\
4 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
-3 \\
-4 \\
-4 \\
-1
\end{array}\right] + x_{4} \left[\begin{array}{c}
1 \\
0 \\
8 \\
0
\end{array}\right] + x_{5} \left[\begin{array}{c}
-5 \\
4 \\
0 \\
-3
\end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\)</article>
</li>
<li id="li-178">The set of vectors \(\left\{ \left[\begin{array}{c}
2 \\
2 \\
0 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-4 \\
4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
8 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
4 \\
0 \\
-3
\end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-30"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-251">Consider the statement <article class="claim theorem-like" id="claim-28"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.16</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
3
\end{array}\right] , \left[\begin{array}{c}
4 \\
-3 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-2 \\
-2 \\
-5
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-5 \\
1 \\
3
\end{array}\right] , \left[\begin{array}{c}
1 \\
4 \\
-2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
1 \\
2
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article></p>
<ol class="lower-alpha">
<li id="li-179">Write an equivalent statement using a vector equation.</li>
<li id="li-180">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-30" id="answer-30"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-30"><div class="answer solution-like">
<div class="displaymath" id="p-252">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccccc}
-1 & 4 & 3 & -2 & 1 & -2 \\
-1 & -3 & -2 & -5 & 4 & 1 \\
-3 & -5 & -2 & 1 & -2 & 1 \\
3 & 0 & -5 & 3 & -3 & 2
\end{array}\right] = \left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & \frac{135}{196} & \frac{1}{4} \\
0 & 1 & 0 & 0 & -\frac{75}{196} & -\frac{1}{4} \\
0 & 0 & 1 & 0 & \frac{93}{196} & -\frac{1}{4} \\
0 & 0 & 0 & 1 & -\frac{44}{49} & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-181">The statement <article class="claim theorem-like" id="claim-29"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.17</span><span class="period">.</span>
</h6>\(\left\{ \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
3
\end{array}\right] , \left[\begin{array}{c}
4 \\
-3 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-2 \\
-2 \\
-5
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-5 \\
1 \\
3
\end{array}\right] , \left[\begin{array}{c}
1 \\
4 \\
-2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
1 \\
2
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)</article> is equivalent to the statement <article class="claim theorem-like" id="claim-30"><h6 class="heading">
<span class="type">Claim</span><span class="space"> </span><span class="codenumber">2.3.18</span><span class="period">.</span>
</h6>\(x_{1} \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
3
\end{array}\right] + x_{2} \left[\begin{array}{c}
4 \\
-3 \\
-5 \\
0
\end{array}\right] + x_{3} \left[\begin{array}{c}
3 \\
-2 \\
-2 \\
-5
\end{array}\right] + x_{4} \left[\begin{array}{c}
-2 \\
-5 \\
1 \\
3
\end{array}\right] + x_{5} \left[\begin{array}{c}
1 \\
4 \\
-2 \\
-3
\end{array}\right] + x_{6} \left[\begin{array}{c}
-2 \\
1 \\
1 \\
2
\end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\)</article>
</li>
<li id="li-182">The set of vectors \(\left\{ \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
3
\end{array}\right] , \left[\begin{array}{c}
4 \\
-3 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-2 \\
-2 \\
-5
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-5 \\
1 \\
3
\end{array}\right] , \left[\begin{array}{c}
1 \\
4 \\
-2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
1 \\
2
\end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)</li>
</ol>
</div></div>
</div></article><p id="p-253"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V3/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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