0004.ptx

<exercise checkit-seed="0004" checkit-slug="A2" checkit-title="Standard matrices">
  <statement>
    <ol>
      <li>Find the standard matrix for the linear transformation <m>S:\mathbb{R}^ 4  \to \mathbb{R}^ 2 </m> given by
              <me>S\left(  \left[\begin{array}{c}
x \\
y \\
z \\
{w}
\end{array}\right]  \right) =  \left[\begin{array}{c}
2 \, x - 3 \, y + z + 6 \, {w} \\
x - y + 2 \, {w}
\end{array}\right] .</me></li>
      <li>Let <m>T:\mathbb{R}^ 4  \to \mathbb{R}^ 3 </m> be the linear transformation given by the standard matrix
              <me> \left[\begin{array}{cccc}
3 &amp; 0 &amp; 4 &amp; 5 \\
0 &amp; 1 &amp; 1 &amp; 3 \\
-1 &amp; -1 &amp; -2 &amp; -4
\end{array}\right] .</me>
              Compute <m>T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right]  \right)</m>.
          </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me> \left[\begin{array}{cccc}
2 &amp; -3 &amp; 1 &amp; 6 \\
1 &amp; -1 &amp; 0 &amp; 2
\end{array}\right] </me>
      </li>
      <li>
        <me>T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right]  \right)= \left[\begin{array}{c}
33 \\
9 \\
-19
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>