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

and

.

**(a)** Determine the basechange matrix

such that

and compute the determinant of

.

A hint was provided as follows: Determine

by specifying its matrix entries

.