What does is mean when a composition of two functions is an isomorphism?

Question

Suppose that $T\colon V \to U$ and $S\colon U\to W$ are linear transformations and that $S\circ T:\colon V\to W$ is an isomorphism. What can we say about the functions $T$ and $S$ in terms of surjectivity, injectivity and bijectivity?

Answer 1

Let $I = S\circ T$. You know that $I$ is linear and an isomorphism by hypothesis, so $\operatorname{ker}(I)= \{0_V\}$. But $\operatorname{ker}T \subset \operatorname{ker} I$ (try to prove this) $\implies T$ is injective. Note that the image of $T(V)$ has dimension $\operatorname{dim} V = \operatorname{dim} W$ since $I$ is an isomorphism. Also, $S$ restricted to $T(V)$ must be injective (otherwise the dimensions of $V$ and $W$ are different) and thus bijective. So $S$ is surjective (and bijective when restricted to $T(V)$).

Note that those are the only facts we can state about the 3 properties, since $T(x,y) = (x,y,0)$ and $S(x,y,z) = (x,y)$ would be counterexamples to other statements (first is injective but not surjective, second is surjective but not injective and their composition is an automorphism). And we’re done.