Example
Let B = {(1,1,0), (1, -1, 0), (0, 0, 1)} and
B' = {(1,0,0), (1,1,0), (-1,0,-1)}. Then B and B' are all bases for $R^3$.
Find the transition matrix $P_{B \to B'}$.
$P_{B \to B'} = \begin{bmatrix} 0 & 2 & -1 \ 1 & -1 & 0 \ 0 & 0 & -1 \end{bmatrix}$.
Find $[(2,0,3)]_{B'}$ using the transition matrix.