$U,V,W$ are finite dimensional vector spaces over a field $\mathbb{F}$. Let $S:U\rightarrow V,\; T:V\rightarrow W$ be linear maps.
Prove: $\text{Ker}(S)\subset\text{Ker}({TS})$
Attempt:
$\text{Ker}({TS})=\left\{ u\in U\;:\;({T\circ S})(u)=0\right\} =\left\{ u\in U\;:\; S( u)=0\;\bigvee\; T({S(u)})=0\right\}
=\left\{ u\in U\;:\; u\in\text{Ker}(S)\;\bigvee\; S(u)\in\text{Im}(S)\cap\text{Ker}(T)\right\}$
$\iff\text{Ker}({TS})=\text{Ker}(S)\cup({\text{Im}(S)\cap\text{Ker}(T))}\supset\text{Ker}(S)$
I was told however that there is a mistake here. What is my mistake?
Thank you.
Best Answer
Let $v\in \ker(S)$. Then $S(v) = 0$, so $T(S(v)) = 0$; hence $v\in \ker(TS)$.