Let V = P5, the vector space of polynomials of degree 5, with coefficients in R, and let W = { p(x) P5 | p(0) = p(1) = p(2) }

1) Show that W is a subspace of V

2)Let u(x) = x(x 1)(x 2), and explain why for every q W there exists r P2 such that q(x) = u(x) r(x) + q(0).

3)Find a basis B for W and order it by increasing degree.

4)For u as in part (2), find the coordinate vector [u]B of u with respect to B from part (3).