This is from Axler's Linear Algebra Done Right: Chapter 3 Question 14:
Suppose that $W$ is finite dimensional and $T \in L(V,W)$ Prove that if $T$ is injective, then there exists $S \in L(W,V)$ such that $ST$ is the identity map on $V$?
I do not understand why $T$ has to be injective?
For example, why can't just define $S\in L(W,V)$ such that $S(Tv) = v$
and then $(ST)v = S(Tv) = v$?