A2-9.ptx

<exercise checkit-seed="0009" 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_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right] \right) = \left[\begin{array}{c}
4 \, x_{1} + x_{2} - 7 \, x_{3} - 6 \, x_{4} \\
3 \, x_{1} + x_{2} - 5 \, x_{3} - 4 \, x_{4}
\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; 0 &amp; -1 &amp; 4
\end{array}\right].</me>
              Compute <m>T\left(\left[\begin{array}{c}
1 \\
-2 \\
7 \\
-6
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{cccc}
4 &amp; 1 &amp; -7 &amp; -6 \\
3 &amp; 1 &amp; -5 &amp; -4
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
1 \\
-2 \\
7 \\
-6
\end{array}\right]\right)=\left[\begin{array}{c}
-30
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>