Q:

$S \circ (T + U) = S \circ T + S \circ U$

Let $S, T$ and $U$ be linear maps $V \longrightarrow V$ .Proving that $(S\circ T)\circ U = S\circ (T\circ U)$and $S \circ (T + U) = S \circ T + S  \circ U$ will require what series of steps?What is the trick here?

A:

The sum of two maps is defined element-wise, 
$$(S+T)\vec v:= S\vec v+ T\vec v$$
where $+$ in RHS is the sum of vectors defined in $V$.
For every vector $\vec v\in V$, you get
$$S(T+U)v=S(T\vec v +U\vec v)=ST\vec v+ SU\vec v=(ST+SU)\vec v$$
Associativity holds in general for functions, not only for linear maps.

