0010.ptx

<exercise checkit-seed="0010" checkit-slug="A4" checkit-title="Injectivity and surjectivity">
  <statement>
      Let <m>T:\mathbb{R}^ 4  \to \mathbb{R}^ 4 </m> be the linear transformation given by the standard matrix
      <m> \left[\begin{array}{cccc}
1 &amp; 2 &amp; -1 &amp; 3 \\
0 &amp; 1 &amp; -2 &amp; 7 \\
0 &amp; 0 &amp; 1 &amp; -3 \\
0 &amp; 1 &amp; 0 &amp; 1
\end{array}\right] .</m><ol><li>Explain why <m>T</m> is or is not injective.</li><li>Explain why <m>T</m> is or is not surjective.</li></ol></statement>
  <answer>
    <p>
      <me>\operatorname{RREF} \left[\begin{array}{cccc}
1 &amp; 2 &amp; -1 &amp; 3 \\
0 &amp; 1 &amp; -2 &amp; 7 \\
0 &amp; 0 &amp; 1 &amp; -3 \\
0 &amp; 1 &amp; 0 &amp; 1
\end{array}\right] = \left[\begin{array}{cccc}
1 &amp; 0 &amp; 0 &amp; -2 \\
0 &amp; 1 &amp; 0 &amp; 1 \\
0 &amp; 0 &amp; 1 &amp; -3 \\
0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right] </me>
    </p>
    <ol>
      <li><m>T</m> is not injective</li>
      <li><m>T</m> is not surjective</li>
    </ol>
  </answer>
</exercise>