A2-8.ptx

<exercise checkit-seed="0008" 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}
z \\
-x - 3 \, y + 6 \, z + 3 \, {w}
\end{array}\right].</me>
          </li>
          <li>Let <m>T:\mathbb{R}^3 \to \mathbb{R}^3</m>
              be the linear transformation given by the standard matrix
              <me>\left[\begin{array}{ccc}
-5 &amp; -2 &amp; -3 \\
-2 &amp; -1 &amp; -2 \\
0 &amp; -2 &amp; -7
\end{array}\right].</me>
              Compute <m>T\left(\left[\begin{array}{c}
3 \\
-4 \\
-6
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{cccc}
0 &amp; 0 &amp; 1 &amp; 0 \\
-1 &amp; -3 &amp; 6 &amp; 3
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
3 \\
-4 \\
-6
\end{array}\right]\right)=\left[\begin{array}{c}
11 \\
10 \\
50
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>