\begin{exercise}{G4}{Eigenvectors}{0007}
\begin{exerciseStatement}
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix \[ \left[\begin{array}{cccc}
4 & -2 & -1 & -1 \\
-2 & 7 & 2 & 2 \\
1 & -2 & 2 & -1 \\
4 & -8 & -4 & -1
\end{array}\right] \]
\end{exerciseStatement}
\begin{exerciseAnswer}
\[\operatorname{RREF} \left[\begin{array}{cccc}
1 & -2 & -1 & -1 \\
-2 & 4 & 2 & 2 \\
1 & -2 & -1 & -1 \\
4 & -8 & -4 & -4
\end{array}\right] = \left[\begin{array}{cccc}
1 & -2 & -1 & -1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c}
2 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
0 \\
1
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
