Let be a field and let be a positive natural number. Denote by the set of all polynomials having degree at most . Fix a nonzero . Let and let . Compute the transition matrix . Conclude that is a basis for .
Using the binomial theorem, we have . Letting these coordinates (in ) be the entries of the th column of , we see that where if and 0 otherwise. In particular, is upper triangular and has all 1s on the main diagonal- hence this matrix is invertible. Thus is a basis for .