So I’ve been stuck on this question for some time now and I can’t seem to get the ball rolling. It is clear to me that to prove either $T$ is onto or $S$ one-to-one, I must find whether if $n\geq m$. Nevertheless, I’m stumped on the next step to take to demonstrate this. Perhaps you might be able to help me out. 

Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. 

Suppose $AB = I\_{m}$, that is the $m\times m$ identity matrix. 

Consider the linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $S: \mathbb{R}^m\rightarrow \mathbb{R}^n$ respectively defined by $[T] = A$ and $[S]= B$.

(a) Show that $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is onto.

(b) Show that $S:\mathbb{R}^m\rightarrow\mathbb{R}^n$ is one-to-one.