Proposition (Isomorphism $\iff$ bijective)

A Linear Map $T\in\mathscr{L}(V,W)$, is an Isomorphism if and only if $T$ is a bijection.

Proposition (Injectivity, surjectivity, and isomorphism are equivalent when dimension is the same)

If $V,W$ are finite dimensional vector spaces such that $dim(V)=dim(W)$, and we have some Linear Map $T\in\mathscr{L}(V,W)$, then the following are equivalent:

1. $T$ is Injective
2. $T$ is Surjective
3. $T$ is Invertible