Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & 2 & 7 & 2 \\ 2 & -3 & -8 & 4 \\ 0 & 3 & 10 & 0 \\ -2 & 6 & 6 & 0 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 7 & 2 \\ 2 & -7 & -8 & 4 \\ 0 & 3 & 6 & 0 \\ -2 & 6 & 6 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 3 & 2 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -3 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).