A2-1.ptx

<exercise checkit-seed="0001" checkit-slug="A2" checkit-title="Standard matrices">
  <statement>
      <ol>
          <li>Find the standard matrix for the linear transformation
              <m>S:\mathbb{R}^3 \to \mathbb{R}^4</m> given by
              <me>S\left( \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \right) = \left[\begin{array}{c}
-x + 3 \, y + 4 \, z \\
-x - z \\
2 \, x - 2 \, y - z \\
-3 \, x + y - 6 \, z
\end{array}\right].</me>
          </li>
          <li>Let <m>T:\mathbb{R}^4 \to \mathbb{R}^2</m>
              be the linear transformation given by the standard matrix
              <me>\left[\begin{array}{cccc}
2 &amp; 6 &amp; 1 &amp; -3 \\
1 &amp; 3 &amp; 1 &amp; -2
\end{array}\right].</me>
              Compute <m>T\left(\left[\begin{array}{c}
-3 \\
-7 \\
6 \\
-5
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{ccc}
-1 &amp; 3 &amp; 4 \\
-1 &amp; 0 &amp; -1 \\
2 &amp; -2 &amp; -1 \\
-3 &amp; 1 &amp; -6
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
-3 \\
-7 \\
6 \\
-5
\end{array}\right]\right)=\left[\begin{array}{c}
-27 \\
-8
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>