1
<exercise checkit-seed="0008" checkit-slug="A4" checkit-title="Injectivity and surjectivity">
2
  <statement>
3
      <p>
4
        Let <m>T:\mathbb{R}^5 \to \mathbb{R}^3</m> be the linear transformation given by the standard matrix
5
        <m>\left[\begin{array}{ccccc}
6
5 &amp; -3 &amp; 3 &amp; 25 &amp; 12 \\
7
4 &amp; -3 &amp; 1 &amp; 17 &amp; 13 \\
8
-2 &amp; 1 &amp; -2 &amp; -12 &amp; -3
9
\end{array}\right]</m>.
10
      </p>
11
      <ol>
12
            <li><p>Explain why <m>T</m> is or is not injective.</p></li>
13
            <li><p>Explain why <m>T</m> is or is not surjective.</p></li>
14
      </ol>
15
  </statement>
16
  <answer>
17
      <p><me>\operatorname{RREF}\left[\begin{array}{ccccc}
18
5 &amp; -3 &amp; 3 &amp; 25 &amp; 12 \\
19
4 &amp; -3 &amp; 1 &amp; 17 &amp; 13 \\
20
-2 &amp; 1 &amp; -2 &amp; -12 &amp; -3
21
\end{array}\right]=\left[\begin{array}{ccccc}
22
1 &amp; 0 &amp; 0 &amp; 2 &amp; 3 \\
23
0 &amp; 1 &amp; 0 &amp; -2 &amp; -1 \\
24
0 &amp; 0 &amp; 1 &amp; 3 &amp; -2
25
\end{array}\right]</me></p>
26
      <ol>
27
          <li>
28
              <p><m>T</m> is not  injective.</p>
29
          </li>
30
          <li>
31
              <p><m>T</m> is  surjective.</p>
32
          </li>
33
      </ol>
34
  </answer>
35
</exercise>
36

37