<exercise checkit-seed="0003" checkit-slug="A4" checkit-title="Injectivity and surjectivity">
<statement>
<p>
Let <m>T:\mathbb{R}^4 \to \mathbb{R}^5</m> be the linear transformation given by the standard matrix
<m>\left[\begin{array}{cccc}
1 & 0 & -2 & 3 \\
0 & 1 & 4 & -1 \\
0 & 0 & 1 & 0 \\
0 & -1 & -5 & 1 \\
2 & 2 & -3 & 4
\end{array}\right]</m>.
</p>
<ol>
<li><p>Explain why <m>T</m> is or is not injective.</p></li>
<li><p>Explain why <m>T</m> is or is not surjective.</p></li>
</ol>
</statement>
<answer>
<p><me>\operatorname{RREF}\left[\begin{array}{cccc}
1 & 0 & -2 & 3 \\
0 & 1 & 4 & -1 \\
0 & 0 & 1 & 0 \\
0 & -1 & -5 & 1 \\
2 & 2 & -3 & 4
\end{array}\right]=\left[\begin{array}{cccc}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]</me></p>
<ol>
<li>
<p><m>T</m> is not injective.</p>
</li>
<li>
<p><m>T</m> is not surjective.</p>
</li>
</ol>
</answer>
</exercise>
