Problem 5. Consider the vector space Pn (a) Verify that (5.9) = [5(0)9(x)V1 22 de is an inner product on Pn. (b) Show that for this inner product, (gh, f) = (h,gf) for all f, g, h E Pn. (c) Let (po(2), p.(2), p2(2), ..., Pr (2)) be the orthonormal basis of Pn obtained by applying the Gram-Schmidt process to (1, 1, 12,..., 2"). Given that (1, 1) = 7/2, (, 2) = 7/8, and (z?, ?) 7/16, compute po(3), p1(2), p2(2). (Hint: The integral is easy for odd functions! Also, remember (b).] (d) Show that pn is orthogonal in this inner product to rk for all k 2: IPn-1(x) = anPn(x) + an-1Pn-2(x) where ak (P#(2), xpk-1(2)). [HINT: Write xpn-1 as a linear combination of your orthonormal basis. What are the coefficients? Again, make use of (b). Problem 5. Consider the vector space Pn (a) Verify that (5.9) = [5(0)9(x)V1 22 de is an inner product on Pn. (b) Show that for this inner product, (gh, f) = (h,gf) for all f, g, h E Pn. (c) Let (po(2), p.(2), p2(2), ..., Pr (2)) be the orthonormal basis of Pn obtained by applying the Gram-Schmidt process to (1, 1, 12,..., 2"). Given that (1, 1) = 7/2, (, 2) = 7/8, and (z?, ?) 7/16, compute po(3), p1(2), p2(2). (Hint: The integral is easy for odd functions! Also, remember (b).] (d) Show that pn is orthogonal in this inner product to rk for all k 2: IPn-1(x) = anPn(x) + an-1Pn-2(x) where ak (P#(2), xpk-1(2)). [HINT: Write xpn-1 as a linear combination of your orthonormal basis. What are the coefficients? Again, make use of (b)