Problem 1PP: Use the inner product axioms to verify the following statements. 1. v,0=0,v=0. Problem 2PP: Use the inner product axioms to verify the following statements. 2. u,v+w=u,v+u,w Problem 1E: Let 2 have the inner product of Example 1, and let x = (1, 1) and y = (5, 1). a. Find x, y, and... Problem 2E: Let 2 have the inner product of Example 1. Show that the Cauchy-Schwarz inequality holds for x = (3,... Problem 3E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 4E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 5E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 6E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 7E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 8E: Exercises 3-8 refer to 2 with the inner product given by evaluation at 1, 0, and 1. (See Example 2.)... Problem 9E: Let 3 have the inner product given by evaluation at 3, 1, 1, and 3. Let p0(t) = 1, p1(t) = t, and... Problem 10E: Let 3 have the inner product as in Exercise 9, with p0, p1, and q the polynomials described there.... Problem 11E: Let p0, p1, and p2 be the orthogonal polynomials described in Example 5, where the inner product on... Problem 12E: Find a polynomial p3 such that {p0, p1, p2, p3} (see Exercise 11) is an orthogonal basis for the... Problem 13E: Let A be any invertible n n matrix. Show that for u, v in n, the formula u,v = (Au) (Av) =... Problem 14E: Let T be a one-to-one linear transformation from a vector space V into n. Show that for u, v in V,... Problem 15E: Use the inner product axioms and other results of this section to verify the statements in Exercises... Problem 16E: Use the inner product axioms and other results of this section to verify the statements in Exercises... Problem 17E: Use the inner product axioms and other results of this section to verify the statements in Exercises... Problem 18E: Use the inner product axioms and other results of this section to verify the statements in Exercises... Problem 19E: Given a 0 and b 0, let u=[ab] and v=[ba]. Use the Cauchy-Schwarz inequality to compare the... Problem 20E: Let u=[ab] and v=[11]. Use the Cauchy-Schwarz inequality to show that (a+b2)2a2+b22 Problem 21E: Exercises 21-24 refer to V = C[0, 1], with the inner product given by an integral, as in Example 7.... Problem 22E: Exercises 21-24 refer to V = C[0, 1], with the inner product given by an integral, as in Example 7.... Problem 23E: Compute f for f in Exercise 21. Exercises 21-24 refer to V = C[0, 1], with the inner product given... Problem 24E: Compute g for g in Exercise 22. Exercises 21-24 refer to V = C[0, 1], with the inner product given... Problem 25E: Let V be the space C[1, 1] with the inner product of Example 7. Find an orthogonal basis for the... Problem 26E: Let V be the space C[2, 2] with the inner product of Example 7. Find an orthogonal basis for the... format_list_bulleted