Composition of Linear Maps is Linear

Theorem

• If $T:U\to V$ is linear
• If $S:V\to W$ is linear
• Then $ST:U\to W$ is also linear

Proof

1. Suppose $T$ and $S$ are both linear
2. Suppose the composition $ST:U\to W$ is defined
3. Pick $x,y\in U$ and $a,b,\in \mathbb{F}$
4. We calculate:
   1. $ST(ax+by)$
   2. $=S(aT(x)+bT(y))$
   3. $=aS(T(x))+bS(T(x))$a
   4. …