\begin{exercise}{G4}{Eigenvectors}{0010}
\begin{exerciseStatement}
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix \[ \left[\begin{array}{cccc}
4 & 2 & 0 & -5 \\
1 & 6 & 1 & -8 \\
0 & 2 & 5 & -6 \\
0 & 0 & 0 & 3
\end{array}\right] \]
\end{exerciseStatement}
\begin{exerciseAnswer}
\[\operatorname{RREF} \left[\begin{array}{cccc}
1 & 2 & 0 & -5 \\
1 & 3 & 1 & -8 \\
0 & 2 & 2 & -6 \\
0 & 0 & 0 & 0
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & -2 & 1 \\
0 & 1 & 1 & -3 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c}
2 \\
-1 \\
1 \\
0
\end{array}\right] , \left[\begin{array}{c}
-1 \\
3 \\
0 \\
1
\end{array}\right] \right\} \).
\end{exerciseAnswer}
\end{exercise}
