Let's see if you can follow me: any element in can be seen as a change of basis transformation, since a linear map is invertible iff it maps a basis into a basis, so what the problem is giving is that T is a linear map s. t. (why?).

What's left now is to prove that the only lin. maps that commute with ALL the elements in L(V) are the scalar multiples of the identity map, and this is a nice, fairly non-hard exercise. One way to approach it is to work with matrices...

Tonio