[Math] Show that (i) $\dim\ker T=\dim\ker ST$ (ii) $\dim\operatorname{im}T=\dim\operatorname{im}ST$

Question

Let $T: V \rightarrow W$ be a linear mapping and $S: W \rightarrow U$ be an isomorphism (all finite dimensional vector spaces). Show that

(i) $\dim\ker T=\dim\ker ST$

(ii) $\dim\operatorname{im}T=\dim\operatorname{im} ST$

Attempt;

Clearly, $T: V \rightarrow W$; $S: W \rightarrow U$ then $ST:V\rightarrow U$

Also $\dim\ker T+\dim\operatorname{im} T=n=\dim V$

$\dim\operatorname{im}S=m=\dim W$ as $\dim\ker S=0$, for $S$ is isomorphism

and $\dim\ker ST+\dim\operatorname{im}ST=n=\dim V$

what to do next?

Answer 1

$\newcommand{\im}{\operatorname{im}}$I think the easiest way to do this would be to directly show $\ker T=\ker ST$ (which I encourage you do to on your own; of course $\ker T\subseteq\ker ST$ is trivial so you just need to do the reverse). With this in mind, using rank-nullity we have

$$\dim V=\dim\ker T+\dim\im T$$ and $$\dim V=\dim\ker ST+\dim\im ST$$

However, since you know $\dim\ker T=\dim\ker ST$, you can set the two equations equal and recover $\dim\im T=\dim\im ST$.