Matrix Multiplication of Composite Maps

Definition

1. With bases $\alpha =\{u_1,\dots ,u_n\}$
2. With bases $\beta =\{v_1,\dots ,v_k\}$
3. With basis $\gamma =\{w_1,\dots ,w_t\}$
4. If $S:V\to V$
5. If $T:V\to W$
6. Then, the product $[T{]}_{\beta }^{\gamma }\ast [S{]}_{\alpha }^{\gamma }=[TS{]}_{\alpha }^{\gamma }$