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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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<span class="type">Section</span> <span class="codenumber">2.4</span> <span class="title">Subspaces (V4)</span>
</h2>
<article class="definition definition-like" id="definition-12"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.4.1</span><span class="period">.</span>
</h6>
<p id="p-254">A subset of a vector space is called a <dfn class="terminology">subspace</dfn> if it is a vector space on its own.</p>
<p id="p-255">For example, the span of these two vectors forms a planar subspace inside of the larger vector space \(\IR^3\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{center}
\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);
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\end{center}
</pre></article><article class="fact theorem-like" id="fact-5"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.4.2</span><span class="period">.</span>
</h6>
<p id="p-256">Any sub<em class="emphasis">set</em> \(S\) of a vector space \(V\) that contains the additive identity \(\vec 0\) satisfies the eight vector space properties automatically, since it is a collection of known vectors.</p>
<p id="p-257">However, to verify that it's a sub<em class="emphasis">space</em>, we need to check that addition and multiplication still make sense using only vectors from \(S\text{.}\) So we need to check two things:</p>
<ul class="disc">
<li id="li-183"><p id="p-258">The set is <dfn class="terminology">closed under addition</dfn>: for any \(\vec{x},\vec{y} \in S\text{,}\) the sum \(\vec{x}+\vec{y}\) is also in \(S\text{.}\)</p></li>
<li id="li-184"><p id="p-259">The set is <dfn class="terminology">closed under scalar multiplication</dfn>: for any \(\vec{x} \in S\) and scalar \(c \in \IR\text{,}\) the product \(c\vec{x}\) is also in \(S\text{.}\)</p></li>
</ul></article><article class="activity project-like" id="activity-40"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.4.1</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-17"><p id="p-260">Let \(S=\setBuilder{\left[\begin{array}{c} x \\ y \\ z \end{array}\right]}{ x+2y+z=0}\text{.}\)</p></div>
<article class="task exercise-like" id="task-39"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-261">Let \(\vec{v}=\left[\begin{array}{c} x \\ y \\ z \end{array}\right]\) and \(\vec{w} = \left[\begin{array}{c} a \\ b \\ c \end{array}\right] \) be vectors in \(S\text{,}\) so \(x+2y+z=0\) and \(a+2b+c=0\text{.}\) Show that \(\vec v+\vec w = \left[\begin{array}{c} x+a \\ y+b \\ z+c \end{array}\right]\) also belongs to \(S\) by verifying that \((x+a)+2(y+b)+(z+c)=0\text{.}\)</p></article><article class="task exercise-like" id="task-40"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-262">Let \(\vec{v}=\left[\begin{array}{c} x \\ y \\ z \end{array}\right]\in S\text{,}\) so \(x+2y+z=0\text{.}\) Show that \(c\vec v=\left[\begin{array}{c}cx\\cy\\cz\end{array}\right]\) also belongs to \(S\) for any \(c\in\IR\) by verifying an appropriate equation.</p></article><article class="task exercise-like" id="task-41"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-263">Is \(S\) is a subspace of \(\IR^3\text{?}\)</p></article></article><article class="activity project-like" id="activity-41"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.4.2</span><span class="period">.</span>
</h6>
<p id="p-264">Let \(S=\setBuilder{\left[\begin{array}{c} x \\ y \\ z \end{array}\right]}{ x+2y+z=4}\text{.}\) Choose a vector \(\vec v=\left[\begin{array}{c} \unknown\\\unknown\\\unknown \end{array}\right]\) in \(S\) and a real number \(c=\unknown\text{,}\) and show that \(c\vec v\) isn't in \(S\text{.}\) Is \(S\) a subspace of \(\IR^3\text{?}\)</p></article><article class="remark remark-like" id="remark-11"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">2.4.3</span><span class="period">.</span>
</h6>
<p id="p-265">Since \(0\) is a scalar and \(0\vec{v}=\vec{z}\) for any vector \(\vec{v}\text{,}\) a nonempty set that is closed under scalar multiplication must contain the zero vector \(\vec{z}\) for that vector space.</p>
<p id="p-266">Put another way, you can check any of the following to show that a nonempty subset \(W\) isn't a subspace:</p>
<ul class="disc">
<li id="li-185"><p id="p-267">Show that \(\vec 0\not\in W\text{.}\)</p></li>
<li id="li-186"><p id="p-268">Find \(\vec u,\vec v\in W\) such that \(\vec u+\vec v\not\in W\text{.}\)</p></li>
<li id="li-187"><p id="p-269">Find \(c\in\IR,\vec v\in W\) such that \(c\vec v\not\in W\text{.}\)</p></li>
</ul>
<p id="p-270">If you cannot do any of these, then \(W\) can be proven to be a subspace by doing the following:</p>
<ul class="disc">
<li id="li-188"><p id="p-271">Prove that \(\vec u+\vec v\in W\) whenever \(\vec u,\vec v\in W\text{.}\)</p></li>
<li id="li-189"><p id="p-272">Prove that \(c\vec v\in W\) whenever \(c\in\IR,\vec v\in W\text{.}\)</p></li>
</ul></article><article class="activity project-like" id="activity-42"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.4.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-18">
<p id="p-273">Consider these subsets of \(\IR^3\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
R=
\setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{y=z+1}
\hspace{2em}
S=
\setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{y=|z|}
\hspace{2em}
T=
\setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{z=xy}\text{.}
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-42"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-274">Show \(R\) isn't a subspace by showing that \(\vec 0\not\in R\text{.}\)</p></article><article class="task exercise-like" id="task-43"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-275">Show \(S\) isn't a subspace by finding two vectors \(\vec u,\vec v\in S\) such that \(\vec u+\vec v\not\in S\text{.}\)</p></article><article class="task exercise-like" id="task-44"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-276">Show \(T\) isn't a subspace by finding a vector \(\vec v\in T\) such that \(2\vec v\not\in T\text{.}\)</p></article></article><article class="activity project-like" id="activity-43"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">2.4.4</span><span class="period">.</span>
</h6>
<p id="p-277">Let \(W\) be a subspace of a vector space \(V\text{.}\) How are \(\vspan W\) and \(W\) related?</p>
<ol class="lower-alpha">
<li id="li-190"><p id="p-278">\(\vspan W\) is bigger than \(W\)</p></li>
<li id="li-191"><p id="p-279">\(\vspan W\) is the same as \(W\)</p></li>
<li id="li-192"><p id="p-280">\(\vspan W\) is smaller than \(W\)</p></li>
</ol></article><article class="fact theorem-like" id="fact-6"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">2.4.4</span><span class="period">.</span>
</h6>
<p id="p-281">If \(S\) is any subset of a vector space \(V\text{,}\) then since \(\vspan S\) collects all possible linear combinations, \(\vspan S\) is automatically a subspace of \(V\text{.}\)</p>
<p id="p-282">In fact, \(\vspan S\) is always the smallest subspace of \(V\) that contains all the vectors in \(S\text{.}\)</p></article><section class="exercises" id="exercises-7"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">2.4.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-31"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-283">Consider the following two sets of Euclidean vectors:</p>
<div class="displaymath">
\begin{align*}
\left\{ \left[\begin{array}{c}
x \\
y
\end{array}\right] \middle|\,x^{3} + 7 \, y = 0\right\} & & \left\{ \left[\begin{array}{c}
x \\
y
\end{array}\right] \middle|\,7 \, x + 5 \, y = 0\right\}
\end{align*}
</div>
<p id="p-284">Explain why one of these sets is a subspace of \(\mathbb{R}^ 2 \) and one is not.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-31" id="answer-31"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-31"><div class="answer solution-like"><p id="p-285">\(U\) is a subspace of \(\mathbb{R}^ 2 \) and \(W\) is not.</p></div></div>
</div></article><article class="exercise exercise-like" id="exercise-32"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-286">Consider the following two sets of Euclidean vectors:</p>
<div class="displaymath">
\begin{align*}
\left\{ \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \middle|\,6 \, x = 5 \, y + 2 \, z\right\} & & \left\{ \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \middle|\,y^{2} + 4 \, x = 4 \, z\right\}
\end{align*}
</div>
<p id="p-287">Explain why one of these sets is a subspace of \(\mathbb{R}^ 3 \) and one is not.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-32" id="answer-32"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-32"><div class="answer solution-like"><p id="p-288">\(U\) is a subspace of \(\mathbb{R}^ 3 \) and \(W\) is not.</p></div></div>
</div></article><article class="exercise exercise-like" id="exercise-33"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-289">Consider the following two sets of Euclidean vectors:</p>
<div class="displaymath">
\begin{align*}
\left\{ \left[\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right] \middle|\,x^{2} + 5 \, y + 2 \, z = 4 \, w\right\} & & \left\{ \left[\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right] \middle|\,3 \, w + 3 \, y = 6 \, x - 5 \, z\right\}
\end{align*}
</div>
<p id="p-290">Explain why one of these sets is a subspace of \(\mathbb{R}^ 4 \) and one is not.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-33" id="answer-33"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-33"><div class="answer solution-like"><p id="p-291">\(U\) is a subspace of \(\mathbb{R}^ 4 \) and \(W\) is not.</p></div></div>
</div></article><article class="exercise exercise-like" id="exercise-34"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-292">Consider the following two sets of Euclidean vectors:</p>
<div class="displaymath">
\begin{align*}
\left\{ \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \middle|\,3 \, x + 7 \, y = 2 \, z\right\} & & \left\{ \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \middle|\,y^{3} + 6 \, x = 3 \, z\right\}
\end{align*}
</div>
<p id="p-293">Explain why one of these sets is a subspace of \(\mathbb{R}^ 3 \) and one is not.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-34" id="answer-34"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-34"><div class="answer solution-like"><p id="p-294">\(U\) is a subspace of \(\mathbb{R}^ 3 \) and \(W\) is not.</p></div></div>
</div></article><article class="exercise exercise-like" id="exercise-35"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-295">Consider the following two sets of Euclidean vectors:</p>
<div class="displaymath">
\begin{align*}
\left\{ \left[\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right] \middle|\,7 \, x^{3} y + 5 \, w z = 0\right\} & & \left\{ \left[\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right] \middle|\,3 \, w + y = x - 5 \, z\right\}
\end{align*}
</div>
<p id="p-296">Explain why one of these sets is a subspace of \(\mathbb{R}^ 4 \) and one is not.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-35" id="answer-35"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-35"><div class="answer solution-like"><p id="p-297">\(U\) is a subspace of \(\mathbb{R}^ 4 \) and \(W\) is not.</p></div></div>
</div></article><p id="p-298"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/V4/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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