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