Chapter 7.1, Problem 47E

Eigenvalues and Eigenvectors of Linear Transformations In Exercises 45-48, consider the linear transformation $T$: $R^n\to R^n$whose matrix $A$relative to the standard basis is given. Find (a) the eigenvalues of $A$, (b) a basis for each of the corresponding eigenspaces, and (c) the matrix $A\text{'}$for $T$relative to the basis $B\text{'}$, where $B\text{'}$is made up of the basis vectors found in part (b).

$\left[\begin{array}{ccc}0& 2& -1\\ -1& 3& 1\\ 0& 0& -1\end{array}\right]$