- 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.

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.