0007.ptx

<exercise checkit-seed="0007" checkit-slug="G4" checkit-title="Eigenvectors">
  <statement>
    <p>Explain how to find a basis for the eigenspace associated to the eigenvalue <m> 3 </m> in the matrix <me> \left[\begin{array}{cccc}
4 &amp; -2 &amp; -1 &amp; -1 \\
-2 &amp; 7 &amp; 2 &amp; 2 \\
1 &amp; -2 &amp; 2 &amp; -1 \\
4 &amp; -8 &amp; -4 &amp; -1
\end{array}\right] </me></p>
  </statement>
  <answer>
    <p>
      <me>\operatorname{RREF} \left[\begin{array}{cccc}
1 &amp; -2 &amp; -1 &amp; -1 \\
-2 &amp; 4 &amp; 2 &amp; 2 \\
1 &amp; -2 &amp; -1 &amp; -1 \\
4 &amp; -8 &amp; -4 &amp; -4
\end{array}\right] = \left[\begin{array}{cccc}
1 &amp; -2 &amp; -1 &amp; -1 \\
0 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right] </me>
    </p>
    <p>A basis of the eigenspace is <m> \left\{ \left[\begin{array}{c}
2 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
0 \\
1
\end{array}\right] \right\} </m>.</p>
  </answer>
</exercise>