0012.ptx

<exercise checkit-seed="0012" 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_{1} \\
x_{2} \\
x_{3}
\end{array}\right]  \right) =  \left[\begin{array}{c}
4 \, x_{1} + x_{2} + 8 \, x_{3} \\
-x_{1} - 2 \, x_{3} \\
x_{1} + 3 \, x_{2} + 3 \, x_{3} \\
-2 \, x_{1} + 3 \, x_{2} - 8 \, x_{3}
\end{array}\right] .</me></li>
      <li>Let <m>T:\mathbb{R}^ 4  \to \mathbb{R}^ 3 </m> be the linear transformation given by the standard matrix
              <me> \left[\begin{array}{cccc}
-1 &amp; -5 &amp; -1 &amp; -4 \\
-2 &amp; -5 &amp; 2 &amp; -7 \\
1 &amp; 4 &amp; 0 &amp; 4
\end{array}\right] .</me>
              Compute <m>T\left( \left[\begin{array}{c}
-3 \\
7 \\
-1 \\
0
\end{array}\right]  \right)</m>.
          </li>
    </ol>
  </statement>
  <answer>
    <ol>
      <li>
        <me> \left[\begin{array}{ccc}
4 &amp; 1 &amp; 8 \\
-1 &amp; 0 &amp; -2 \\
1 &amp; 3 &amp; 3 \\
-2 &amp; 3 &amp; -8
\end{array}\right] </me>
      </li>
      <li>
        <me>T\left( \left[\begin{array}{c}
-3 \\
7 \\
-1 \\
0
\end{array}\right]  \right)= \left[\begin{array}{c}
-31 \\
-31 \\
25
\end{array}\right] </me>
      </li>
    </ol>
  </answer>
</exercise>