0006.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>