Understanding bases for eigenspaces of a matrix

Solution 1:

So you have the matrix$$\begin{bmatrix}-8&3&-8\\0&0&0\\1&0&1\end{bmatrix}.$$Then the eigenvectors of $A$ corresponding to the eigenvalue $-2$ are the non-zero vectors $(x_1,x_2,x_3)$ such that$$\begin{bmatrix}-8&3&-8\\0&0&0\\1&0&1\end{bmatrix}.\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}.$$This is the same thing as asserting that$$\left\{\begin{array}{l}-8x_1+3x_2+8x_3=0\\x_1+x_3=0,\end{array}\right.$$which is equivalent to $x_3=-x_1$ and $x_2=0$. So, the eigenvectors that you are interested in are those of the form$$\begin{bmatrix}x_1\\0\\-x_1\end{bmatrix}=x_1\begin{bmatrix}1\\0\\-1\end{bmatrix}$$and so a basis of the eigenspace corresponding to the eigenvalue $-2$ is$$\begin{bmatrix}1\\0\\-1\end{bmatrix}.$$