A2-4.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>