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