I'd be immensely grateful for any help with this problem:
Let be a finite-dimensional vector space, and let
be a linear operator (a linear map) such that , for every linear map .
Prove that there exists a scalar such that .
Oh, and there has been a hint provided:
Hint: show that has at least one eigenvalue and observe the corresponding eigenspace.
First, how could I show that has at least one eigenvalue? Of course, if has an eigenvalue , then there exists a vector , , such that ; but how to show that there is such an eigenvalue in the first place?
And I would really appreciate if someone could show me how to proceed from that to the final solution.