Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ be given by $T(x, y, z) = (x - y, y - z)$. Let $B = \{(1, 1, 1), (1, 1, 0), (0, 1, 1)\}$ and $B' = \{(1, 1), (2, 1)\}$. Let $v = (2, 4, 6)$.
a) Find the matrix for $T$ relative to $B$ and $B'$.
b) Find $T(v)$ using the matrix relative to $B$ and $B'$.