Let $B = \{(1, 1, 0), (0, 1, 1), (1, 0, 1)\}$ and $B' = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ be bases for $\mathbb{R}^3$, and let
$A = \begin{bmatrix} \frac{1}{2} & \frac{-1}{2} & \frac{5}{2} \\ 2 & 1 & 2 \\ \frac{-1}{2} & \frac{3}{2} & \frac{1}{2} \end{bmatrix}$
be the matrix for $T: \mathbb{R}^3 \to \mathbb{R}^3$ relative to $B$.
(a) Find the transition matrix $P$ from $B'$ to $B$.
(b) Use the matrices $P$ and $A$ to find $[v]_B$ and $[T(v)]_B$, where
$[v]_{B'} = \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}$.
