Express the dimensions of the null space and range space of $S$ and $T$ in terms of $\dim V , \dim W$

Question

I need to solve the following problem.

Let $V$ and $W$ be finite dimensional vector spaces and let $S\colon V \to W$ and $T\colon W \to V$ be linear maps such that $ST=I_v ={\rm identity}$. Express the dimensions of the null space and range space of $S$ and $T$ in terms of $\dim V, \dim W$.

My attempt

Let $\dim V=n $ and $\dim W=m$. Since $ST=I_v$, nullity of $S=0$. Thus $\operatorname{rank}(S)=\dim W=m$.

But how do I find that of $T$ ?

Answer 1

Since $ST=\operatorname{Id}$, $\ker T=\{0\}$ and, in particular, $\dim\ker T=0$. So, $\dim T(W)=\dim W$.

On the other hand, since $ST=\operatorname{Id}$, $S(V)=W$ and therefore $\dim S(V)=\dim W$. And then, by the rank-nullity theorem,$$\dim\ker S=\dim V-\dim S(V)=\dim V-\dim W.$$