can you give me the full explanation process for the problem (b), I can not understand the solution very well.

Give P3(R) the inner product (p, q) = p(x)q(r) dr. (a) Find an orthonormal basis of the subspace U = {p : p(1) = 0} CP3(R). (b) Find an orthonormal basis for U-, and an orthonormal basis for P3 (R) that extends your orthonormal basis for U. (c) Find the polynomial p E P3 (R) such that p(1) = 0 and (1 + 3x - p(x)) dx is as small as possible.(2) If we extend our orthonormal basis e1, e2, es to an orthonormal basis of P;(R), then the new basis vector es will be orthogonal to all the others, and so it will be a basis of the orthogonal complement UL. First we extend our orthonormal basis of U to a basis of P;(R) by adding a fourth vector va = 1 which is not in U. Then we use Gram-Schmidt to determine a vector e, which makes an orthonormal basis e1, e2, e3, es of P3(R). MATH 436 HOMEWORK 11-SOLUTIONS 3 es = WA - (vA, el)el - (vA, e2)e2 - (vA, e3) e3 =1 - (1,x - 1) . 3(x - 1) - (1, 4x2 - 5x + 1) . 5(4x] - 5x + 1) - (1, 15x3- 25x' + 11x - 1) . 7(15x3- 25x' + 11x - 1) =1+(x- 1)+ 2(4x2 - 5x + 1)+ 15, (15x3 - 25x2 + 11x - 1) = =(35x3 - 45x2 + 15x - 1) lleAll = = eA = 35x3 - 45x' + 15x - 1. So, ex is an orthonormal basis for US, and e1, e2, e3, ex is an orthonormal basis for P;(R) extending the orthonormal basis of U