Determinant of linear transform with ordered basis is independent of the choice

Given a basis for define . Prove that the definition is independent of choice of an ordered basis for V (i.e. if and are ordered basis for then ).

I don't have much in the form of a proof. I tried working out an example but it got me nowhere. So what I have as an example is:

let and then under the transformation we have

where

and for

where

where was obtained by and which gives the desired result.

Now I'm thinking it has something with the eigenvalues and corresponding eigenvectors but that got nowhere, since I'm not to sure on how to proceeds.