0011.ptx

<exercise checkit-seed="0011" checkit-slug="A3" checkit-title="Image and kernel">
  <statement>
      Let <m>T:\mathbb{R}^ 3  \to \mathbb{R}^ 4 </m> be the linear transformation given by
      <me>T\left(  \left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]  \right) =  \left[\begin{array}{c}
x_{1} - x_{2} + 2 \, x_{3} \\
x_{2} - 2 \, x_{3} \\
x_{1} - x_{2} + 2 \, x_{3} \\
-x_{2} + 2 \, x_{3}
\end{array}\right] .</me><ol><li>Explain how to find the image of <m>T</m> and the kernel of <m>T</m>.</li><li>Explain how to find a basis of the image of <m>T</m> and a basis of the kernel of <m>T</m>.</li><li>Explain how to find the rank and nullity of <m>T</m>, and why the rank-nullity theorem holds for <m>T</m>.</li></ol></statement>
  <answer>
    <p>
      <me>\operatorname{RREF} \left[\begin{array}{ccc}
1 &amp; -1 &amp; 2 \\
0 &amp; 1 &amp; -2 \\
1 &amp; -1 &amp; 2 \\
0 &amp; -1 &amp; 2
\end{array}\right] = \left[\begin{array}{ccc}
1 &amp; 0 &amp; 0 \\
0 &amp; 1 &amp; -2 \\
0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0
\end{array}\right] </me>
    </p>
    <ol>
      <li>
        <me>\operatorname{Im}\ T = \operatorname{span}\  \left\{ \left[\begin{array}{c}
1 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-1 \\
1 \\
-1 \\
-1
\end{array}\right] \right\} </me>
        <me>\operatorname{ker}\ T =  \left\{ \left[\begin{array}{c}
0 \\
2 \, a \\
a
\end{array}\right] \middle|\,a\in\mathbb{R}\right\} </me>
      </li>
      <li>
              A basis of <m>\operatorname{Im}\ T</m> is <m> \left\{ \left[\begin{array}{c}
1 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-1 \\
1 \\
-1 \\
-1
\end{array}\right] \right\} </m>.
              A basis of <m>\operatorname{ker}\ T</m> is <m> \left\{ \left[\begin{array}{c}
0 \\
2 \\
1
\end{array}\right] \right\} </m></li>
      <li>
              The rank of <m>T</m> is <m> 2 </m>, the nullity of <m>T</m> is <m> 1 </m>, 
              and the dimension of the domain of <m>T</m> is <m> 3 </m>. The rank-nullity theorem asserts that
              <m> 2 + 1 = 3 </m>, which we see to be true.
          </li>
    </ol>
  </answer>
</exercise>