A2-3.ptx

<exercise checkit-seed="0003" 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}^3</m> given by
              <me>S\left( \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right] \right) = \left[\begin{array}{c}
x \\
y - 4 \, z \\
-y + 5 \, z
\end{array}\right].</me>
          </li>
          <li>Let <m>T:\mathbb{R}^2 \to \mathbb{R}^3</m>
              be the linear transformation given by the standard matrix
              <me>\left[\begin{array}{cc}
-5 &amp; 3 \\
3 &amp; -2 \\
0 &amp; 2
\end{array}\right].</me>
              Compute <m>T\left(\left[\begin{array}{c}
-5 \\
3
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{ccc}
1 &amp; 0 &amp; 0 \\
0 &amp; 1 &amp; -4 \\
0 &amp; -1 &amp; 5
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
-5 \\
3
\end{array}\right]\right)=\left[\begin{array}{c}
34 \\
-21 \\
6
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>