I Prove that if T is injective, T*T is invertible

[JJ__]

TL;DR Summary

Suppose V, W are inner product spaces, and V is finite dimensional. I need to prove that if T: V-->W is an injective linear map, then T*T is invertible.

I'm using the notation T* to indicate the adjoint of T.

I got as far as to say that if T is injective, then T* is surjective. But I don't know how to show that T*T is invertible. Showing that T*T is surjective or injective would imply invertibility, but I'm not sure how to do that either. I was hoping to find a way to show that T* is injective (which would then imply that T*T is injective) but I wasn't able to.

 

[black hole 123]

im pretty rusty on this so double check for mistakes... to show T*T is injective u just need to show if T*Tu=0, then u=0.

0=<T*Tu,u>=<Tu,Tu>, so Tu=0. since T is injective, u=0

T*T is linear and injective and goes from V->V, so range of T*T must have same dimension as V, i.e. range of T*T is V itself, i.e. its surjective. looks like its important for V to be finite dimensional since this wouldn't apply if its infinite...