Note: For this question to display nicely, you may need to close the menu on the left side of the eClass page. Let B = {1 + 3x + 5r', 2 + 52 + 1012, 3 + 8x + 1613} CP2 . (a) Prove that B is a basis for P2. (@) Show that B is linearly independent. Suppose a(1 + 3: + 5x?) + b(2 + 5x + 10r?) + c(3 + 8x + 161?) = 0 for some a, b, ce IR. We need to show that a = 0, b = 0, c = 0. Now a(1 + 3x + 5r?) + b(2 + 5x + 10z?) + c(3 + 8 + 161?) = 0 p 1+ d pitd z? =0 = (0)1 + (0)z +(0)13 = 0 (by equating coefficients of = 0 1, I and 12, respectively) = 0 This gives us the augmented matrix Tref E Therefore, the only solution is a = 0, b = 0 and c = 0. Thus a(1 + 3x + 5x?) + 6(2 + 51 + 10r?) + c(3 + 81 + 161?) = 0 a=b = c=0, so B is linearly independent. (ii) To show that the set B spans Pa we must show that for any f + gr + ha?