Linear Map Composition

Definition

1. Given two functions $T:U\to V$ and $S:V->W$
2. Their composition $ST:U\to W$ is given by $ST(x)=S(T(x))=S\circ T$

Properties

Associative

If we have 3 linear maps: 1. $R:U\to V$ 2. $S:V\to W$ 3. $T:W\to X$ Then, $(TS)R=T(SR)$ Linear Map Composition Associative Proof

Distributive

If we have 3 linear maps: 1. $R:U\to V$ 2. $S:V\to W$ 3. $T:W\to X$ Then, $(T+S)R=TR+SR$ Linear Map Composition Distributive Proof

Theorems

• Composition of Linear Maps is Linear
• Kernels Images and Compositions Theorem
• Matrix Multiplication
• Composition of Linear Maps are not Commutative