1. Composition of linear maps

Suppose $U:$   R$^n \rightarrow$   R$^m$ and $T:$   R$^m \rightarrow$   R$^k$ are linear maps. Then it makes sense to consider the composition $T \circ U$ given by $(T \circ U)($x$) = T(U($x$))$. Notice it's $U$ that is applied first. In terms of the matrices, if $U = T _ A$ and $T = T _ B$, then $(T \circ U)($x$) = (T _ A \circ T _ B) ($x$) = A(B$x$) = (AB)($x$) = T _ {AB}($x$)$. Thus composition of linear maps corresponds to matrix multiplication. The order of the matrices is the same as the order of the maps when written as a composition.

This explains in particular why matrix multiplication is associative, i.e., why $(AB)C = A(BC)$: Composition is obviously associative, since $(S \circ T) \circ U$ and $S \circ (T \circ U)$ applied to x are both just $S(T(U($x$)))$.