Coordination System Change Matrix

Use the following general result: if $B=\{e_1,e_2,e_3\}$ and $B'=\{e'_1,e'_2,e'_3\}$ are basis of $\mathbb{R}^3$ and $e'_i=\sum_{j=1}^3{a_{ij}e_j}$ for $i=1,2,3$ , then $[v]_B=\begin{bmatrix}{a_{11}}&{a_{21}}&{a_{31}}\\{a_{ 12}}&{a_{22}}&{a_{32}}\\{a_{13}}&{a_{23}}&{a_{33}} \end{bmatrix}[v]_{B'}$