I'm having trouble showing this:

Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective (respectively, surjective).

Just as an aside, is there an "induces" arrow?

I'm having trouble showing this:

Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective (respectively, surjective).

I've been using the dimension theorem to show relatedness through cardinality, but this has unearthed nothing. I was thinking that a method using canonical isomorphisms might be in order. I suspect the answer is just an $iff$-chain of theorems.

Just as an aside, is there an "induces" arrow?