Assume $(V, \langle{}\cdot{},{}\cdot{}\rangle)_v$ and $(W, \langle {}\cdot{},{}\cdot{}\rangle)_w$ are finite dimensional inner product spaces. $T:V \rightarrow W$. Prove that $T^*T\colon V \rightarrow V$ is bijective.
I know how to prove the injection, but I am stuck on the surjection. Can I get a sketch of the proof or an idea of what I should look out for?