It's possible to show that $-2$ is an eigenvalue for $\left[\begin{array}{ccc}-1&4&-2\\2&-7&9\\3&0&4\end{array}\right]\text{.}$

Compute the kernel of the transformation with standard matrix

Since the kernel of a linear map is a subspace of $\IR^n\text{,}$ and the kernel obtained from $A-\lambda I$ contains all the eigenvectors associated with $\lambda\text{,}$ we call this kernel the eigenspace of $A$ associated with $\lambda\text{.}$

Find a basis for the eigenspace for the matrix $\left[\begin{array}{ccc} 0 & 0 & 3 \\ 1 & 0 & -1 \\ 0 & 1 & 3 \end{array}\right]$ associated with the eigenvalue $3\text{.}$

Find a basis for the eigenspace for the matrix $\left[\begin{array}{cccc} 5 & -2 & 0 & 4 \\ 6 & -2 & 1 & 5 \\ -2 & 1 & 2 & -3 \\ 4 & 5 & -3 & 6 \end{array}\right]$ associated with the eigenvalue $1\text{.}$

Find a basis for the eigenspace for the matrix $\left[\begin{array}{cccc} 4 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 0 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2 \end{array}\right]$ associated with the eigenvalue $2\text{.}$