Composition of linear maps

Definition

Let $F$ be a field, let $U$, $V$, $W$ be vector spaces over $F$, and let $S \colon U \to V$ and $T \colon V \to W$ be linear maps. The composition of $T$ and $S$ is the map $$T \circ S \colon U \to W, \qquad (T \circ S)(u) = T(S(u)).$$

Remarks

The composition of linear maps is linear. Composition is associative, and the identity maps are its identity elements. The matrix of a composition is the product of the matrices, in the same order. Composition is not commutative: in general $T \circ S \ne S \circ T$, even when both are defined.

Explanations

L1 human

To compose two linear maps, we apply one and then the other. We write $T \circ S$ for "first $S$, then $T$". Matrix multiplication is exactly this operation, written in coordinates.