Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 4 & -2 & -1 & 5 \\ 0 & 4 & -2 & 1 \\ -1 & 2 & 5 & -6 \\ 0 & 1 & -7 & 9 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ -1 & 2 & 2 & -6 \\ 0 & 1 & -7 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 1 \end{array}\right] \right\} \).