(a) Consider the linear transformation $\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by the matrix

$$\left(\begin{array}{rrr} 5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{array}\right)$$

Find a basis of $\mathbb{R}^{3}$ in which $\alpha$ is represented by a diagonal matrix.

(b) Give a list of $6 \times 6$ matrices such that any linear transformation $\beta: \mathbb{R}^{6} \rightarrow \mathbb{R}^{6}$ with characteristic polynomial

$$(x-2)^{4}(x+7)^{2}$$

and minimal polynomial

$$(x-2)^{2}(x+7)$$

is similar to one of the matrices on your list. No two distinct matrices on your list should be similar. [No proof is required.]