0009.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>