0002.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>