0001.tex

\begin{exercise}{G4}{Eigenvectors}{0001}
\begin{exerciseStatement}

Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix \[ \left[\begin{array}{cccc}
4 & 8 & 1 & -4 \\
2 & 10 & -1 & -4 \\
1 & 4 & -1 & -2 \\
-1 & -4 & -2 & 4
\end{array}\right] \]

\end{exerciseStatement}
\begin{exerciseAnswer}

\[\operatorname{RREF} \left[\begin{array}{cccc}
2 & 8 & 1 & -4 \\
2 & 8 & -1 & -4 \\
1 & 4 & -3 & -2 \\
-1 & -4 & -2 & 2
\end{array}\right] = \left[\begin{array}{cccc}
1 & 4 & 0 & -2 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \]



A basis of the eigenspace is \( \left\{ \left[\begin{array}{c}
-4 \\
1 \\
0 \\
0
\end{array}\right] , \left[\begin{array}{c}
2 \\
0 \\
0 \\
1
\end{array}\right] \right\} \).

\end{exerciseAnswer}
\end{exercise}