Kernels Images and Compositions Theorem

Theorem

1. Suppose $S:U\to V$ is a linear map
2. Suppose $T:V\to W$ is a linear map
3. Then, $ker(S)\subset ker(TS)$
4. Then, $Image(TS)\subset Image(T)$

Proof

Proving $ker(S)\subset ker(TS)$

1. Pick $v\in Ker(S)$
2. Calculate $TS(v)=T(S(v))$ by defn of $TS$
3. $=T(0_v)$ as $v\in Ker(S)$
4. $=0_w$ as $T$ is linear
5. Therefore, $v\in Ker(TS)$ as it follows $Ker(S)=Ker(TS)$

Proving $Image(TS)\subset Image(T)$

1. Pick $w\in Image(TS)$
2. $w=TS(u)$ by defn of Image
3. $=T(S(u))$ by defn of $TS$
4. $=T(v)$ as $S(u)=V$
5. Therefore, $w\in Image(T)$S