Some important facts about composition of linear maps.

Suppose we have three linear mapping $F,G,H$ s.t. $F: V \rightarrow U$ and $G: U \rightarrow W$ and $H: W \rightarrow Z$.

We have

$F \circ ( G \circ H) $ = $ (F \circ G) \circ H$, i.e. the associative law. This property is on the textbook (Serge Lang, Linear Algebra 3rd edition) page 47

and

$F \circ (G + H ) = F \circ G + F \circ H$, i.e. the distributive law. See page 67 for more details.

However, the commutative law does not apply in general. In some special cases, it is applicable, though. For example, $F \circ F = F \circ F = F^2$.