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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="A4"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">3.4</span> <span class="title">Injective and Surjective Linear Maps (A4)</span>
</h2>
<article class="definition definition-like" id="definition-22"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.4.1</span><span class="period">.</span>
</h6>
<p id="p-593">Let \(T: V \rightarrow W\) be a linear transformation. \(T\) is called <dfn class="terminology">injective</dfn> or <dfn class="terminology">one-to-one</dfn> if \(T\) does not map two distinct vectors to the same place. More precisely, \(T\) is injective if \(T(\vec{v}) \neq T(\vec{w})\) whenever \(\vec{v} \neq \vec{w}\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1)
node[anchor=south east] {\(\vec v\)};
\draw[thick,-latex,red] (0,0) -- (1.5,-2)
node[anchor=south west] {\(\vec w\)};
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1)
node[anchor=south west] {\(T(\vec v)\)};
\draw[thick,-latex,red] (0,0) -- (1,2)
node[anchor=south west] {\(T(\vec w)\)};
\end{scope}
\node[anchor=north] at (5,-1) {injective};
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1)
node[anchor=south west] {\(\vec v\)};
\draw[thick,-latex,red] (0,0) -- (1,2)
node[anchor=south west] {\(\vec w\)};
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,purple] (0,0) -- (0.5,2)
node[anchor=south west] {\(T(\vec v)=T(\vec w)\)};
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\node[anchor=north] at (5,-1) {not injective};
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</pre></article><article class="activity project-like" id="activity-90"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.1</span><span class="period">.</span>
</h6>
<p id="p-594">Let \(T: \IR^3 \rightarrow \IR^2\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c}x \\ y\\z \end{array}\right] \right)
=
\left[\begin{array}{c} x \\ y \end{array}\right]
\hspace{3em}
\text{with standard matrix }
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Is \(T\) injective?</p>
<ol class="lower-alpha">
<li id="li-430"><p id="p-595">Yes, because \(T(\vec v)=T(\vec w)\) whenever \(\vec v=\vec w\text{.}\)</p></li>
<li id="li-431"><p id="p-596">Yes, because \(T(\vec v)\not=T(\vec w)\) whenever \(\vec v\not=\vec w\text{.}\)</p></li>
<li id="li-432"><p id="p-597">No, because \(T\left(\left[\begin{array}{c}0\\0\\1\end{array}\right]\right)
\not=
T\left(\left[\begin{array}{c}0\\0\\2\end{array}\right]\right)\)</p></li>
<li id="li-433"><p id="p-598">No, because \(T\left(\left[\begin{array}{c}0\\0\\1\end{array}\right]\right)
=
T\left(\left[\begin{array}{c}0\\0\\2\end{array}\right]\right)\)</p></li>
</ol></article><article class="activity project-like" id="activity-91"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.2</span><span class="period">.</span>
</h6>
<p id="p-599">Let \(T: \IR^2 \rightarrow \IR^3\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c}x \\ y \end{array}\right] \right)
=
\left[\begin{array}{c} x \\ y \\ 0 \end{array}\right]
\hspace{3em}
\text{with standard matrix }
\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Is \(T\) injective?</p>
<ol class="lower-alpha">
<li id="li-434"><p id="p-600">Yes, because \(T(\vec v)=T(\vec w)\) whenever \(\vec v=\vec w\text{.}\)</p></li>
<li id="li-435"><p id="p-601">Yes, because \(T(\vec v)\not=T(\vec w)\) whenever \(\vec v\not=\vec w\text{.}\)</p></li>
<li id="li-436"><p id="p-602">No, because \(T\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)
\not=
T\left(\left[\begin{array}{c}3\\4\end{array}\right]\right)\)</p></li>
<li id="li-437"><p id="p-603">No, because \(T\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)
=
T\left(\left[\begin{array}{c}3\\4\end{array}\right]\right)\)</p></li>
</ol></article><article class="definition definition-like" id="definition-23"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.4.2</span><span class="period">.</span>
</h6>
<p id="p-604">Let \(T: V \rightarrow W\) be a linear transformation. \(T\) is called <dfn class="terminology">surjective</dfn> or <dfn class="terminology">onto</dfn> if every element of \(W\) is mapped to by an element of \(V\text{.}\) More precisely, for every \(\vec{w} \in W\text{,}\) there is some \(\vec{v} \in V\) with \(T(\vec{v})=\vec{w}\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (1,2);
\draw[thick,-latex,blue] (0,0) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (1,-1);
\draw[thick,-latex,blue] (0,0) -- (-2,3);
\draw[thick,-latex,blue] (0,0) -- (-3,-2);
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (0.5,2);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1);
\draw[thick,-latex,blue] (0,0) -- (0,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2,-2);
\fill[color=blue, opacity=0.5] (-2,-2) rectangle (2,2);
\end{scope}
\node[anchor=north] at (5,-2) {surjective};
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (0.5,2);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1);
\draw[thick,-latex,blue] (0,0) -- (0,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2,-2);
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (0.5,1.5);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-2.5,1);
\draw[thick,-latex,blue] (0,0) -- (-0.5,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2.5,-0.5);
\fill[color=blue, opacity=0.5] (-2,-2) -- (3,-1) -- (2,2) -- (-3,1) -- (-2,-2);
\end{scope}
\node[anchor=north] at (5,-2) {not surjective};
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-92"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.3</span><span class="period">.</span>
</h6>
<p id="p-605">Let \(T: \IR^2 \rightarrow \IR^3\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c}x \\ y \end{array}\right] \right)
=
\left[\begin{array}{c} x \\ y \\ 0 \end{array}\right]
\hspace{3em}
\text{with standard matrix }
\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Is \(T\) surjective?</p>
<ol class="lower-alpha">
<li id="li-438"><p id="p-606">Yes, because for every \(\vec w=\left[\begin{array}{c}x\\y\\z\end{array}\right]\in\IR^3\text{,}\) there exists \(\vec v=\left[\begin{array}{c}x\\y\end{array}\right]\in\IR^2\) such that \(T(\vec v)=\vec w\text{.}\)</p></li>
<li id="li-439"><p id="p-607">No, because \(T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)\) can never equal \(\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]
\text{.}\)</p></li>
<li id="li-440"><p id="p-608">No, because \(T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)\) can never equal \(\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]
\text{.}\)</p></li>
</ol></article><article class="activity project-like" id="activity-93"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.4</span><span class="period">.</span>
</h6>
<p id="p-609">Let \(T: \IR^3 \rightarrow \IR^2\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c}x \\ y\\z \end{array}\right] \right)
=
\left[\begin{array}{c} x \\ y \end{array}\right]
\hspace{3em}
\text{with standard matrix }
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">Is \(T\) surjective?</p>
<ol class="lower-alpha">
<li id="li-441"><p id="p-610">Yes, because for every \(\vec w=\left[\begin{array}{c}x\\y\end{array}\right]\in\IR^2\text{,}\) there exists \(\vec v=\left[\begin{array}{c}x\\y\\42\end{array}\right]\in\IR^3\) such that \(T(\vec v)=\vec w\text{.}\)</p></li>
<li id="li-442"><p id="p-611">Yes, because for every \(\vec w=\left[\begin{array}{c}x\\y\end{array}\right]\in\IR^2\text{,}\) there exists \(\vec v=\left[\begin{array}{c}0\\0\\z\end{array}\right]\in\IR^3\) such that \(T(\vec v)=\vec w\text{.}\)</p></li>
<li id="li-443"><p id="p-612">No, because \(T\left(\left[\begin{array}{c}x\\y\\z\end{array}\right]\right)\) can never equal \(\left[\begin{array}{c} 3\\-2 \end{array}\right]
\text{.}\)</p></li>
</ol></article><article class="observation remark-like" id="observation-19"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">3.4.3</span><span class="period">.</span>
</h6>
<p id="p-613">As we will see, it's no coincidence that the \(\RREF\) of the injective map's standard matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">has all pivot columns. Similarly, the \(\RREF\) of the surjective map's standard matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]
\end{equation*}
</div>
<p data-braille="continuation">has a pivot in each row.</p></article><article class="activity project-like" id="activity-94"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.5</span><span class="period">.</span>
</h6>
<p id="p-614">Let \(T: V \rightarrow W\) be a linear transformation where \(\ker T\) contains multiple vectors. What can you conclude?</p>
<ol class="lower-alpha">
<li id="li-444"><p id="p-615">\(T\) is injective</p></li>
<li id="li-445"><p id="p-616">\(T\) is not injective</p></li>
<li id="li-446"><p id="p-617">\(T\) is surjective</p></li>
<li id="li-447"><p id="p-618">\(T\) is not surjective</p></li>
</ol></article><article class="fact theorem-like" id="fact-18"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">3.4.4</span><span class="period">.</span>
</h6>
<p id="p-619">A linear transformation \(T\) is injective <em class="emphasis">if and only if</em> \(\ker T = \{\vec{0}\}\text{.}\) Put another way, an injective linear transformation may be recognized by its <dfn class="terminology">trivial</dfn> kernel.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1)
node[anchor=south east] {\(\vec v\)};
\draw[thick,-latex,red] (0,0) -- (1.5,-2)
node[anchor=south west] {\(\vec w\)};
\fill[green!50!black] (0,0) circle (0.2)
node[anchor=south west] {\(\vec{0}\)};
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1)
node[anchor=south west] {\(T(\vec v)\)};
\draw[thick,-latex,red] (0,0) -- (1,2)
node[anchor=south west] {\(T(\vec w)\)};
\fill[green!50!black] (0,0) circle (0.2)
node[anchor=south east] {\(T(\vec{0})=\vec{0}\)};
\end{scope}
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-95"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.6</span><span class="period">.</span>
</h6>
<p id="p-620">Let \(T: V \rightarrow \IR^5\) be a linear transformation where \(\Im T\) is spanned by four vectors. What can you conclude?</p>
<ol class="lower-alpha">
<li id="li-448"><p id="p-621">\(T\) is injective</p></li>
<li id="li-449"><p id="p-622">\(T\) is not injective</p></li>
<li id="li-450"><p id="p-623">\(T\) is surjective</p></li>
<li id="li-451"><p id="p-624">\(T\) is not surjective</p></li>
</ol></article><article class="fact theorem-like" id="fact-19"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">3.4.5</span><span class="period">.</span>
</h6>
<p id="p-625">A linear transformation \(T:V \rightarrow W\) is surjective <em class="emphasis">if and only if</em> \(\Im T = W\text{.}\) Put another way, a surjective linear transformation may be recognized by its identical codomain and image.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (1,2);
\draw[thick,-latex,blue] (0,0) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (1,-1);
\draw[thick,-latex,blue] (0,0) -- (-2,3);
\draw[thick,-latex,blue] (0,0) -- (-3,-2);
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (0.5,2);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1);
\draw[thick,-latex,blue] (0,0) -- (0,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2,-2);
\fill[color=blue, opacity=0.5] (-2,-2) rectangle (2,2);
\end{scope}
\node[anchor=north] at (5,-2) {surjective, \(\Im T=\IR^2\)};
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.15in,y=0.15in]
\begin{scope}[shift={(0,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (0.5,2);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1);
\draw[thick,-latex,blue] (0,0) -- (0,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2,-2);
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (0.5,1.5);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-2.5,1);
\draw[thick,-latex,blue] (0,0) -- (-0.5,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2.5,-0.5);
\fill[color=blue, opacity=0.5] (-2,-2) -- (3,-1) -- (2,2) -- (-3,1) -- (-2,-2);
\end{scope}
\node[anchor=north] at (5,-2) {not surjective, \(\Im T\not=\IR^3\)};
\end{tikzpicture}
</pre></article><article class="activity project-like" id="activity-96"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.7</span><span class="period">.</span>
</h6>
<p id="p-626">Let \(T: \IR^n \rightarrow \IR^m\) be a linear map with standard matrix \(A\text{.}\) Sort the following claims into two groups of \textit{equivalent} statements: one group that means \(T\) is <em class="emphasis">injective</em>, and one group that means \(T\) is <em class="emphasis">surjective</em>.</p>
<ol class="lower-alpha">
<li id="li-452"><p id="p-627">The kernel of \(T\) is trivial, i.e. \(\ker T=\{\vec 0\}\text{.}\)</p></li>
<li id="li-453"><p id="p-628">The columns of \(A\) span \(\IR^m\text{.}\)</p></li>
<li id="li-454"><p id="p-629">The columns of \(A\) are linearly independent.</p></li>
<li id="li-455"><p id="p-630">Every column of \(\RREF(A)\) has a pivot.</p></li>
<li id="li-456"><p id="p-631">Every row of \(\RREF(A)\) has a pivot.</p></li>
<li id="li-457"><p id="p-632">The image of \(T\) equals its codomain, i.e. \(\Im T=\IR^m\text{.}\)</p></li>
<li id="li-458"><p id="p-633">The system of linear equations given by the augmented matrix \(\left[\begin{array}{c|c}A & \vec{b} \end{array}\right]\) has a solution for all \(\vec{b} \in \IR^m\text{.}\)</p></li>
<li id="li-459"><p id="p-634">The system of linear equations given by the augmented matrix \(\left[\begin{array}{c|c} A & \vec{0} \end{array}\right]\) has exactly one solution.</p></li>
</ol></article><article class="observation remark-like" id="observation-20"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">3.4.6</span><span class="period">.</span>
</h6>
<p id="p-635">The easiest way to determine if the linear map with standard matrix \(A\) is injective is to see if \(\RREF(A)\) has a pivot in each column.</p>
<p id="p-636">The easiest way to determine if the linear map with standard matrix \(A\) is surjective is to see if \(\RREF(A)\) has a pivot in each row.</p></article><article class="activity project-like" id="activity-97"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.8</span><span class="period">.</span>
</h6>
<p id="p-637">What can you conclude about the linear map \(T:\IR^2\to\IR^3\) with standard matrix \(\left[\begin{array}{cc} a & b \\ c & d \\ e & f \end{array}\right]\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-460"><p id="p-638">Its standard matrix has more columns than rows, so \(T\) is not injective.</p></li>
<li id="li-461"><p id="p-639">Its standard matrix has more columns than rows, so \(T\) is injective.</p></li>
<li id="li-462"><p id="p-640">Its standard matrix has more rows than columns, so \(T\) is not surjective.</p></li>
<li id="li-463"><p id="p-641">Its standard matrix has more rows than columns, so \(T\) is surjective.</p></li>
</ol></article><article class="activity project-like" id="activity-98"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.9</span><span class="period">.</span>
</h6>
<p id="p-642">What can you conclude about the linear map \(T:\IR^3\to\IR^2\) with standard matrix \(\left[\begin{array}{ccc} a & b & c \\ d & e & f \end{array}\right]\text{?}\)</p>
<ol class="decimal">
<li id="li-464"><p id="p-643">Its standard matrix has more columns than rows, so \(T\) is not injective.</p></li>
<li id="li-465"><p id="p-644">Its standard matrix has more columns than rows, so \(T\) is injective.</p></li>
<li id="li-466"><p id="p-645">Its standard matrix has more rows than columns, so \(T\) is not surjective.</p></li>
<li id="li-467"><p id="p-646">Its standard matrix has more rows than columns, so \(T\) is surjective.</p></li>
</ol></article><article class="fact theorem-like" id="fact-20"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">3.4.7</span><span class="period">.</span>
</h6>The following are true for any linear map \(T:V\to W\text{:}\) <ul class="disc">
<li id="li-468"><p id="p-647">If \(\dim(V)>\dim(W)\text{,}\) then \(T\) is not injective.</p></li>
<li id="li-469"><p id="p-648">If \(\dim(V)<\dim(W)\text{,}\) then \(T\) is not surjective.</p></li>
</ul>
<p id="p-649">Basically, a linear transformation cannot reduce dimension without collapsing vectors into each other, and a linear transformation cannot increase dimension from its domain to its image.</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[x=0.12in,y=0.12in]
\begin{scope}[shift={(0,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (2,1)
node[anchor=south west] {\(\vec v\)};
\draw[thick,-latex,red] (0,0) -- (1,2)
node[anchor=south west] {\(\vec w\)};
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,purple] (0,0) -- (0.5,2)
node[anchor=south west] {\(T(\vec v)=T(\vec w)\)};
\end{scope}
\node[anchor=north] at (5,-1) {not injective, \(3>2\)};
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.12in,y=0.12in]
\begin{scope}[shift={(0,1)}]
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw[thick,-latex,blue] (0,0) -- (0.5,2);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-1.5,1);
\draw[thick,-latex,blue] (0,0) -- (0,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2,-2);
\end{scope}
\draw[dashed,-latex] (3,3) to [bend left=30] (7,3);
\begin{scope}[shift={(9,0)}]
\draw (0,0) -- (3,0);
\draw (0,0) -- (0,3);
\draw (0,0) -- (-2,-1);
\draw[thick,-latex,blue] (0,0) -- (0.5,1.5);
\draw[thick,-latex,blue] (0,0) -- (2,1);
\draw[thick,-latex,blue] (0,0) -- (-2.5,1);
\draw[thick,-latex,blue] (0,0) -- (-0.5,-1.5);
\draw[thick,-latex,blue] (0,0) -- (2.5,-0.5);
\fill[color=blue, opacity=0.5] (-2,-2) -- (3,-1) -- (2,2) -- (-3,1) -- (-2,-2);
\end{scope}
\node[anchor=north] at (5,-2) {not surjective, \(2<3\)};
\end{tikzpicture}
</pre>
<p id="p-650">But dimension arguments <em class="emphasis">cannot</em> be used to prove a map <em class="emphasis">is</em> injective or surjective.</p></article><article class="activity project-like" id="activity-99"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.10</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-29"><p id="p-651">Suppose \(T: \IR^n \rightarrow \IR^4\) with standard matrix \(A=\left[\begin{array}{cccc}
a_{11}&a_{12}&\cdots&a_{1n}\\
a_{21}&a_{22}&\cdots&a_{2n}\\
a_{31}&a_{32}&\cdots&a_{3n}\\
a_{41}&a_{42}&\cdots&a_{4n}\\
\end{array}\right]\) is both injective and surjective (we call such maps <dfn class="terminology">bijective</dfn>).</p></div>
<article class="task exercise-like" id="task-67"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-652">How many pivot rows must \(\RREF A\) have?</p></article><article class="task exercise-like" id="task-68"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-653">How many pivot columns must \(\RREF A\) have?</p></article><article class="task exercise-like" id="task-69"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-654">What is \(\RREF A\text{?}\)</p></article></article><article class="activity project-like" id="activity-100"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.11</span><span class="period">.</span>
</h6>
<p id="p-655">Let \(T: \IR^n \rightarrow \IR^n\) be a bijective linear map with standard matrix \(A\text{.}\) Label each of the following as true or false.</p>
<ol class="lower-alpha">
<li id="li-470"><p id="p-656">\(\RREF(A)\) is the identity matrix.</p></li>
<li id="li-471"><p id="p-657">The columns of \(A\) form a basis for \(\IR^n\)</p></li>
<li id="li-472"><p id="p-658">The system of linear equations given by the augmented matrix \(\left[\begin{array}{c|c} A & \vec{b} \end{array}\right]\) has exactly one solution for each \(\vec b \in \IR^n\text{.}\)</p></li>
</ol></article><article class="observation remark-like" id="observation-21"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">3.4.8</span><span class="period">.</span>
</h6>
<p id="p-659">The easiest way to show that the linear map with standard matrix \(A\) is bijective is to show that \(\RREF(A)\) is the identity matrix.</p></article><article class="activity project-like" id="activity-101"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.12</span><span class="period">.</span>
</h6>
<p id="p-660">Let \(T: \IR^3 \rightarrow \IR^3\) be given by the standard matrix</p>
<div class="displaymath">
\begin{equation*}
A=\left[\begin{array}{ccc} 2&1&-1 \\ 4&1&1 \\ 6&2&1\end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which of the following must be true?</p>
<ol class="lower-alpha">
<li id="li-473"><p id="p-661">\(T\) is neither injective nor surjective</p></li>
<li id="li-474"><p id="p-662">\(T\) is injective but not surjective</p></li>
<li id="li-475"><p id="p-663">\(T\) is surjective but not injective</p></li>
<li id="li-476"><p id="p-664">\(T\) is bijective.</p></li>
</ol></article><article class="activity project-like" id="activity-102"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.13</span><span class="period">.</span>
</h6>
<p id="p-665">Let \(T: \IR^3 \rightarrow \IR^3\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{ccc} x \\ y \\ z \end{array}\right] \right) =
\left[\begin{array}{c} 2x+y-z \\ 4x+y+z \\ 6x+2y\end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which of the following must be true?</p>
<ol class="lower-alpha">
<li id="li-477"><p id="p-666">\(T\) is neither injective nor surjective</p></li>
<li id="li-478"><p id="p-667">\(T\) is injective but not surjective</p></li>
<li id="li-479"><p id="p-668">\(T\) is surjective but not injective</p></li>
<li id="li-480"><p id="p-669">\(T\) is bijective.</p></li>
</ol></article><article class="activity project-like" id="activity-103"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.14</span><span class="period">.</span>
</h6>
<p id="p-670">Let \(T: \IR^2 \rightarrow \IR^3\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c} x \\ y \end{array}\right] \right) =
\left[\begin{array}{c} 2x+3y \\ x-y \\ x+3y\end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which of the following must be true?</p>
<ol class="lower-alpha">
<li id="li-481"><p id="p-671">\(T\) is neither injective nor surjective</p></li>
<li id="li-482"><p id="p-672">\(T\) is injective but not surjective</p></li>
<li id="li-483"><p id="p-673">\(T\) is surjective but not injective</p></li>
<li id="li-484"><p id="p-674">\(T\) is bijective.</p></li>
</ol></article><article class="activity project-like" id="activity-104"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">3.4.15</span><span class="period">.</span>
</h6>
<p id="p-675">Let \(T: \IR^3 \rightarrow \IR^2\) be given by</p>
<div class="displaymath">
\begin{equation*}
T\left(\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) =
\left[\begin{array}{c} 2x+y-z \\ 4x+y+z\end{array}\right].
\end{equation*}
</div>
<p data-braille="continuation">Which of the following must be true?</p>
<ol class="lower-alpha">
<li id="li-485"><p id="p-676">\(T\) is neither injective nor surjective</p></li>
<li id="li-486"><p id="p-677">\(T\) is injective but not surjective</p></li>
<li id="li-487"><p id="p-678">\(T\) is surjective but not injective</p></li>
<li id="li-488"><p id="p-679">\(T\) is bijective.</p></li>
</ol></article><section class="exercises" id="exercises-16"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">3.4.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-76"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \(\left[\begin{array}{ccc}
1 & 2 & -3 \\
0 & 1 & -3 \\
1 & 1 & 1 \\
1 & 4 & -6
\end{array}\right] .\)<ol class="lower-alpha">
<li id="li-489">Explain why \(T\) is or is not injective.</li>
<li id="li-490">Explain why \(T\) is or is not surjective.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-76" id="answer-76"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-76"><div class="answer solution-like">
<div class="displaymath" id="p-680">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{ccc}
1 & 2 & -3 \\
0 & 1 & -3 \\
1 & 1 & 1 \\
1 & 4 & -6
\end{array}\right] = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-491">\(T\) is injective.</li>
<li id="li-492">\(T\) is not surjective</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-77"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \(\left[\begin{array}{cccc}
2 & 8 & -4 & -4 \\
2 & 8 & -1 & -4 \\
1 & 4 & -3 & -2
\end{array}\right] .\)<ol class="lower-alpha">
<li id="li-493">Explain why \(T\) is or is not injective.</li>
<li id="li-494">Explain why \(T\) is or is not surjective.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-77" id="answer-77"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-77"><div class="answer solution-like">
<div class="displaymath" id="p-681">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
2 & 8 & -4 & -4 \\
2 & 8 & -1 & -4 \\
1 & 4 & -3 & -2
\end{array}\right] = \left[\begin{array}{cccc}
1 & 4 & 0 & -2 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-495">\(T\) is not injective</li>
<li id="li-496">\(T\) is not surjective</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-78"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \(\left[\begin{array}{cccc}
-2 & -3 & 1 & 1 \\
-3 & -5 & 1 & 0 \\
-2 & -5 & 0 & -3
\end{array}\right] .\)<ol class="lower-alpha">
<li id="li-497">Explain why \(T\) is or is not injective.</li>
<li id="li-498">Explain why \(T\) is or is not surjective.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-78" id="answer-78"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-78"><div class="answer solution-like">
<div class="displaymath" id="p-682">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
-2 & -3 & 1 & 1 \\
-3 & -5 & 1 & 0 \\
-2 & -5 & 0 & -3
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 2
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-499">\(T\) is not injective</li>
<li id="li-500">\(T\) is surjective.</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-79"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \(\left[\begin{array}{cccc}
1 & -1 & 0 & 5 \\
0 & 1 & -2 & -1 \\
0 & -4 & 8 & 5 \\
0 & -1 & 2 & -2
\end{array}\right] .\)<ol class="lower-alpha">
<li id="li-501">Explain why \(T\) is or is not injective.</li>
<li id="li-502">Explain why \(T\) is or is not surjective.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-79" id="answer-79"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-79"><div class="answer solution-like">
<div class="displaymath" id="p-683">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
1 & -1 & 0 & 5 \\
0 & 1 & -2 & -1 \\
0 & -4 & 8 & 5 \\
0 & -1 & 2 & -2
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & -2 & 0 \\
0 & 1 & -2 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-503">\(T\) is not injective</li>
<li id="li-504">\(T\) is not surjective</li>
</ol>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-80"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \(\left[\begin{array}{cccc}
1 & -2 & -1 & -8 \\
1 & -1 & 0 & -5 \\
0 & 1 & 2 & 6 \\
-1 & 3 & 0 & 6
\end{array}\right] .\)<ol class="lower-alpha">
<li id="li-505">Explain why \(T\) is or is not injective.</li>
<li id="li-506">Explain why \(T\) is or is not surjective.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-80" id="answer-80"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-80"><div class="answer solution-like">
<div class="displaymath" id="p-684">
\begin{equation*}
\operatorname{RREF} \left[\begin{array}{cccc}
1 & -2 & -1 & -8 \\
1 & -1 & 0 & -5 \\
0 & 1 & 2 & 6 \\
-1 & 3 & 0 & 6
\end{array}\right] = \left[\begin{array}{cccc}
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-507">\(T\) is injective.</li>
<li id="li-508">\(T\) is surjective.</li>
</ol>
</div></div>
</div></article><p id="p-685"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/A4/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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