A2-6.ptx

<exercise checkit-seed="0006" 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_{1} \\
x_{2} \\
x_{3}
\end{array}\right] \right) = \left[\begin{array}{c}
x_{2} + x_{3} \\
-x_{1} - 5 \, x_{3} \\
-x_{1} - 3 \, x_{2} - 7 \, x_{3}
\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}
8 \\
8 \\
8 \\
1
\end{array}\right]\right)</m>.
          </li>
      </ol>
  </statement>
  <answer>
      <ol>
          <li>
              <me>\left[\begin{array}{ccc}
0 &amp; 1 &amp; 1 \\
-1 &amp; 0 &amp; -5 \\
-1 &amp; -3 &amp; -7
\end{array}\right]</me>
          </li>
          <li>
              <me>T\left(\left[\begin{array}{c}
8 \\
8 \\
8 \\
1
\end{array}\right]\right)=\left[\begin{array}{c}
33
\end{array}\right]</me>
          </li>
      </ol>
  </answer>
</exercise>