Q3. For a E F, consider the evaluation map eva: Pn(F) - F defined by eva(p(x)) = p(a). For example, ev3(2 - x + 4x3) = 2 - 3+ 4(3)3. (a) Prove that eva is a linear map. (b) Find a basis for Range(eva) and determine rank(eva). (c) Write down a polynomial p(x) in Ker(eva). No justification required. (d) For any a E F, prove that the set of polynomials UQ = {p E Pn(F) : p(@) = 0} is a subspace of Pn (F). [Hint: Don't work too hard.] (e) Determine dim(Ux). [Hint: If you solved (d) the "easy" way, then you won't have to work too hard here either.]