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