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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="G4"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.4</span> <span class="title">Eigenvectors and Eigenspaces (G4)</span>
</h2>
<article class="activity project-like" id="activity-144"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.4.1</span><span class="period">.</span>
</h6>
<p id="p-958">It's possible to show that \(-2\) is an eigenvalue for \(\left[\begin{array}{ccc}-1&4&-2\\2&-7&9\\3&0&4\end{array}\right]\text{.}\)</p>
<p id="p-959">Compute the kernel of the transformation with standard matrix</p>
<div class="displaymath">
\begin{equation*}
A-(-2)I
=
\left[\begin{array}{ccc} \unknown & 4&-2 \\ 2 & \unknown & 9\\3&0&\unknown \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">to find all the eigenvectors \(\vec x\) such that \(A\vec x=-2\vec x\text{.}\)</p></article><article class="definition definition-like" id="definition-30"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">5.4.1</span><span class="period">.</span>
</h6>
<p id="p-960">Since the kernel of a linear map is a subspace of \(\IR^n\text{,}\) and the kernel obtained from \(A-\lambda I\) contains all the eigenvectors associated with \(\lambda\text{,}\) we call this kernel the <dfn class="terminology">eigenspace</dfn> of \(A\) associated with \(\lambda\text{.}\)</p></article><article class="activity project-like" id="activity-145"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.4.2</span><span class="period">.</span>
</h6>
<p id="p-961">Find a basis for the eigenspace for the matrix \(\left[\begin{array}{ccc}
0 & 0 & 3 \\ 1 & 0 & -1 \\ 0 & 1 & 3
\end{array}\right]\) associated with the eigenvalue \(3\text{.}\)</p></article><article class="activity project-like" id="activity-146"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.4.3</span><span class="period">.</span>
</h6>
<p id="p-962">Find a basis for the eigenspace for the matrix \(\left[\begin{array}{ccc}
5 & -2 & 0 & 4 \\ 6 & -2 & 1 & 5 \\ -2 & 1 & 2 & -3 \\ 4 & 5 & -3 & 6
\end{array}\right]\) associated with the eigenvalue \(1\text{.}\)</p></article><article class="activity project-like" id="activity-147"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.4.4</span><span class="period">.</span>
</h6>
<p id="p-963">Find a basis for the eigenspace for the matrix \(\left[\begin{array}{cccc}
4 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 0 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2
\end{array}\right]\) associated with the eigenvalue \(2\text{.}\)</p></article><section class="exercises" id="exercises-24"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">5.4.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-116"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-964">Explain how to find a basis for the eigenspace associated to the eigenvalue \(-1 \) in the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
-2 & -1 & 0 & 1 \\
-2 & -4 & 4 & 6 \\
-2 & -2 & 0 & 3 \\
-1 & -2 & 3 & 3
\end{array}\right]
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-116" id="answer-116"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-116"><div class="answer solution-like">
<div class="displaymath" id="p-965">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
-1 & -1 & 0 & 1 \\
-2 & -3 & 4 & 6 \\
-2 & -2 & 1 & 3 \\
-1 & -2 & 3 & 4
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-966">A basis of the eigenspace is \(\left\{ \left[\begin{array}{c}
1 \\
0 \\
-1 \\
1
\end{array}\right] \right\} \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-117"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-967">Explain how to find a basis for the eigenspace associated to the eigenvalue \(2 \) in the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
4 & 8 & 1 & -4 \\
2 & 10 & -1 & -4 \\
1 & 4 & -1 & -2 \\
-1 & -4 & -2 & 4
\end{array}\right]
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-117" id="answer-117"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-117"><div class="answer solution-like">
<div class="displaymath" id="p-968">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
2 & 8 & 1 & -4 \\
2 & 8 & -1 & -4 \\
1 & 4 & -3 & -2 \\
-1 & -4 & -2 & 2
\end{array}\right] = \left[\begin{array}{cccc}
1 & 4 & 0 & -2 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-969">A basis of the eigenspace is \(\left\{ \left[\begin{array}{c}
-4 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
2 \\
0 \\
0 \\
1
\end{array}\right] \right\} \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-118"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-970">Explain how to find a basis for the eigenspace associated to the eigenvalue \(1 \) in the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
0 & -5 & -1 & -6 \\
-2 & -4 & -1 & -5 \\
-1 & -4 & 0 & -5 \\
0 & -3 & -1 & -4
\end{array}\right]
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-118" id="answer-118"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-118"><div class="answer solution-like">
<div class="displaymath" id="p-971">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
-1 & -5 & -1 & -6 \\
-2 & -5 & -1 & -5 \\
-1 & -4 & -1 & -5 \\
0 & -3 & -1 & -5
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-972">A basis of the eigenspace is \(\left\{ \left[\begin{array}{c}
1 \\
-1 \\
-2 \\
1
\end{array}\right] \right\} \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-119"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-973">Explain how to find a basis for the eigenspace associated to the eigenvalue \(4 \) in the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
4 & 0 & 0 & 0 \\
-1 & 6 & 0 & 4 \\
0 & 0 & 4 & 0 \\
2 & -4 & 0 & -4
\end{array}\right]
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-119" id="answer-119"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-119"><div class="answer solution-like">
<div class="displaymath" id="p-974">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
0 & 0 & 0 & 0 \\
-1 & 2 & 0 & 4 \\
0 & 0 & 0 & 0 \\
2 & -4 & 0 & -8
\end{array}\right] = \left[\begin{array}{cccc}
1 & -2 & 0 & -4 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-975">A basis of the eigenspace is \(\left\{ \left[\begin{array}{c}
2 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
0 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
4 \\
0 \\
0 \\
1
\end{array}\right] \right\} \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-120"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-976">Explain how to find a basis for the eigenspace associated to the eigenvalue \(3 \) in the matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc}
4 & -2 & -1 & 5 \\
0 & 4 & -2 & 1 \\
-1 & 2 & 5 & -6 \\
0 & 1 & -7 & 9
\end{array}\right]
\end{equation*}
</div>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-120" id="answer-120"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-120"><div class="answer solution-like">
<div class="displaymath" id="p-977">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
1 & -2 & -1 & 5 \\
0 & 1 & -2 & 1 \\
-1 & 2 & 2 & -6 \\
0 & 1 & -7 & 6
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<p id="p-978">A basis of the eigenspace is \(\left\{ \left[\begin{array}{c}
-2 \\
1 \\
1 \\
1
\end{array}\right] \right\} \text{.}\)</p>
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</div></article><p id="p-979"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/G4/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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