Linear Map Composition Distributive Proof

Proof

1. Suppose we have $U\stackrel{R}{\to }V\stackrel{T,S}{\to }W$ linear
2. Check left side:
   1. $(T+S)R$
   2. $=TR+SR$
   3. Picking $u\in U$, calculate:
   4. $[(T+S)R](u)=(T+S)R(u)$ by applying $R$
   5. $=S(R(u))$ by defn of $T+S$
3. Check right side:
   1. $[TR+SR](u)$
   2. $=TR(u)+SR(u)$ by defn of $TR+SR$
   3. $=T(R(u))+S(R(u))$ by applying $TR$ and $SR$
4. Note that L.S = R.S, thus compositions are distributive