A2-2.ptx

<exercise checkit-seed="0002" 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 + y - 2 \, {w} \\
x - y
\end{array}\right].</me>
          </li>
          <li>Let <m>T:\mathbb{R}^4 \to \mathbb{R}^1</m>
              be the linear transformation given by the standard matrix
              <me>\left[\begin{array}{cccc}
1 &amp; -1 &amp; -2 &amp; -1
\end{array}\right].</me>
              Compute <m>T\left(\left[\begin{array}{c}
6 \\
-3 \\
5 \\
-7
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{cccc}
-2 &amp; 1 &amp; 0 &amp; -2 \\
1 &amp; -1 &amp; 0 &amp; 0
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
6 \\
-3 \\
5 \\
-7
\end{array}\right]\right)=\left[\begin{array}{c}
6
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>