tbil-la-A4-3.ptx

<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 &amp; 0 &amp; -2 &amp; 3 \\
0 &amp; 1 &amp; 4 &amp; -1 \\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; -1 &amp; -5 &amp; 1 \\
2 &amp; 2 &amp; -3 &amp; 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 &amp; 0 &amp; -2 &amp; 3 \\
0 &amp; 1 &amp; 4 &amp; -1 \\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; -1 &amp; -5 &amp; 1 \\
2 &amp; 2 &amp; -3 &amp; 4
\end{array}\right]=\left[\begin{array}{cccc}
1 &amp; 0 &amp; 0 &amp; 3 \\
0 &amp; 1 &amp; 0 &amp; -1 \\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 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>