Problem 4. Consider the set V of real polynomials f of degree at most 3 which satisfy f(1) = f(2). Define addition and scalar multiplication of such polynomials by (f + g)(x) = f(x) + g(x) and (af)(x) = a * f(x). (a) Show that V is a vector space over R. [You may assume that the set of functions f : R -> R is a vector space over R with addition and scalar multiplication as above] (b) Show that the following polynomials f1(x) = 1, f2(x) = x^2 - 3x and f3(x) = x^3 - 7x form a basis of V. (c) Is the polynomial g(x) = 3 + x + 2x^2 - x^3 in V? If yes, what are the coordinates of g with respect to the basis in (b)? (d) What is the dimension of V?