Matrix Vector Multiplication for Map and Coordinate Vector

• Given bases $\alpha =\{v_1,\dots ,v_n\}$ of $V$
• $\beta =\{w_1,\dots ,v_k\}$ of $W$
• If $T:V\to W$ is a linear map
• Then, $[T(v){]}_{\beta }=[T{]}_{\alpha }^{\beta }[v{]}_{\alpha }$ which is Matrix Vector Multiplication