0005.ptx

<exercise checkit-seed="0005" checkit-slug="A2" checkit-title="Standard matrices">
  <statement>
    <ol>
      <li>Find the standard matrix for the linear transformation <m>S:\mathbb{R}^ 2  \to \mathbb{R}^ 4 </m> given by
              <me>S\left(  \left[\begin{array}{c}
x \\
y
\end{array}\right]  \right) =  \left[\begin{array}{c}
x - y \\
3 \, x - 2 \, y \\
-3 \, x + 3 \, y \\
-x - y
\end{array}\right] .</me></li>
      <li>Let <m>T:\mathbb{R}^ 3  \to \mathbb{R}^ 4 </m> be the linear transformation given by the standard matrix
              <me> \left[\begin{array}{ccc}
-1 &amp; -1 &amp; 0 \\
-2 &amp; -3 &amp; -7 \\
-2 &amp; -3 &amp; -6 \\
1 &amp; 1 &amp; 5
\end{array}\right] .</me>
              Compute <m>T\left( \left[\begin{array}{c}
7 \\
8 \\
0
\end{array}\right]  \right)</m>.
          </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me> \left[\begin{array}{cc}
1 &amp; -1 \\
3 &amp; -2 \\
-3 &amp; 3 \\
-1 &amp; -1
\end{array}\right] </me>
      </li>
      <li>
        <me>T\left( \left[\begin{array}{c}
7 \\
8 \\
0
\end{array}\right]  \right)= \left[\begin{array}{c}
-15 \\
-38 \\
-38 \\
15
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>