0011.ptx

<exercise checkit-seed="0011" checkit-slug="A2" checkit-title="Standard matrices">
  <statement>
    <ol>
      <li>Find the standard matrix for the linear transformation <m>S:\mathbb{R}^ 3  \to \mathbb{R}^ 2 </m> given by
              <me>S\left(  \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right]  \right) =  \left[\begin{array}{c}
-x + 4 \, y + 2 \, z \\
2 \, x - 8 \, y - 5 \, z
\end{array}\right] .</me></li>
      <li>Let <m>T:\mathbb{R}^ 3  \to \mathbb{R}^ 4 </m> be the linear transformation given by the standard matrix
              <me> \left[\begin{array}{ccc}
0 &amp; -2 &amp; -5 \\
-1 &amp; 6 &amp; 4 \\
0 &amp; -5 &amp; -4 \\
-1 &amp; 3 &amp; -2
\end{array}\right] .</me>
              Compute <m>T\left( \left[\begin{array}{c}
-6 \\
5 \\
4
\end{array}\right]  \right)</m>.
          </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me> \left[\begin{array}{ccc}
-1 &amp; 4 &amp; 2 \\
2 &amp; -8 &amp; -5
\end{array}\right] </me>
      </li>
      <li>
        <me>T\left( \left[\begin{array}{c}
-6 \\
5 \\
4
\end{array}\right]  \right)= \left[\begin{array}{c}
-30 \\
52 \\
-41 \\
13
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>