Proposition (Properties of Linear Maps)

1. Associative: Let $T_1\in\mathscr{L}(V_1,V_2),T_2\in\mathscr{L}(V_2,V_3)$ and $T_3\in\mathscr{L}(V_3,V_4)$. The composition $T_1\circ T_2\circ T_3$ is associative, meaning $$(T_1\circ T_2)\circ T_3 = T_1\circ (T_2\circ T_3)$$
2. Identity: If $T\in\mathscr{L}(V,W)$ then $$I_W\circ T = T = T\circ I_V$$
3. Distributivity: If $S_1\in\mathscr{L}(V_1,V_2),T_1,T_2\in\mathscr{L}(V_2,V_3)$, and $S_2\in \mathscr{L}(V_3,V_4)$ then $$(T_1+T_2)\circ S_1=T_1\circ S_1 + T_2\circ S_1\text{ and } S_2\circ (T_1 + T_2) = S_2\circ T_1 + S_2\circ T_2$$