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).