Let T be a linear operator on a finite-dimensional vector space V, and let β be an ordered basis for V, we define det(T) = det ( [T]β )
i) 1) Prove that det(T – tIv) = det ( [T]β – tI ) for any scalar t and any ordered basis β
ii) 2) Prove that if det (T) ≠ 0 , then T is invertible
it seems to me the first problem boils down to showing that [tIv]β = I, the matrix whose entries are the kronecker delta, no matter what β is, since if so, you can subtract the matrices on the RHS. you may need to invoke the isomorphism between Mn(F) and the elements of End(V) (you need β to exhibit this isomorphism, but the fact that they are isomorphic doesn't depend on β).
The problem defines . In order to see if is well defined we need to prove that the right side does not depends on the chosen basis. Indeed, if is another basis of then, and are similar matrices so, .
Now, fixed an using the standard isomorphism between and we have .