Consider the following question: Let $V$ and $W$ be finite-dimensional vector spaces. Suppose $T: V \rightarrow W$ and $S: W \rightarrow V$ linear maps such that $T \circ S = id_W$. As well, suppose $\dim V = \dim W$. Prove that $S \circ T = id_V$
I'm a bit stuck on this proof. This is what I've done so far.
Take, $w \in W, v \in V$. Since, $T \circ S = id_W$, $T \circ S(w) = id_w \cdot w = w$.
I'm not entirely sure now how to use the fact that $\dim V = \dim W$. I'd like to somehow connect it to injectivity. Then say that since the linear transformations are injective then we have the same mapping to the identity under the composition above. But I'm missing a step to reach this conclusion, or there may be a better way I'm not seeing. Any advice to prove this would be much appreciated!