1. Composition of homogeneous linear transformations

Suppose $T:$   R$^m \rightarrow$   R$^n$ and $U:$   R$^k \rightarrow$   R$^ m$ are h.l.t.'s. Then it makes sense to consider the composition $T \circ U$ given by $(T \circ U)($x$) = T(U($x$))$. In terms of the matrices, if $T = T _ A$ and $U = T _ B$, then $(T \circ U)($x$) = (T _ A \circ T _ B) ($x$) = ($x$B)A =$   x$BA$. Thus composition of h.l.t.'s corresponds to matrix multiplication, but in reverse order¹.

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$)))$.