This is pretty difficult stuff, so be sure to read all of it.
Suppose that and are vector spaces over a field .
is the vector space (over ) of polynomials with coefficients in , i.e.
Fix a positive integer . For this, you may use without proof that the subset of all polynomials of degree or less is a subspace of and that this subspace has two bases
(a) Determine the basechange matrix such that and compute the determinant of .
A hint was provided as follows: Determine by specifying its matrix entries .
(b) Suppose here that and such that for all . Prove that is linearly dependent.
A hint was provided as follows: What is the dimension of ?
I'll try to do some of this, but there's a lot of info that I'm unfamiliar with (primarily the change of basis parts).