Chapter 6.1 Write on paper for upvote. questions: 23, 25, 37, 41 please

10. U = [_3 31. =18 8] 27. a. (2v - w, 3u + 2w) b. llu + vil In Exercises 11-12, find the standard inner product on P2 of the given 28. a. (u - v- 2w, 4u + v) polynomials. b. |12w - vil 11. p = -2+x+ 3x3, q=4-7x2 In Exercises 29-30, sketch the unit circle in R2 using the given inner product. 12. p = -5+ 2x + x , q=3+2x-4x2 29. (u, v) = tu,v, + 184202 30 . ( u , v ) = 21 101 + 2 202 In Exercises 13-14, a weighted Euclidean inner product on R? is In Exercises 31-32, find a weighted Euclidean inner product on R2 given for the vectors u = (u), u2) and v = (v1, U2). Find a matrix that generates it. for which the "unit circle" is the ellipse shown in the accompanying figure. 13. (u, v) = 3u, , + 5u202 14. (u, v) = 4u, V1 + 6u202 31. 32. In Exercises 15-16, a sequence of sample points is given. Use the eval- uation inner product on P; at those sample points to find (p, q) for the polynomials p= x+x and q=1+x2 15. X0 = -2, X1 = -1, x2 = 0, x3 = 1 FIGURE Ex-31 FIGURE Ex-32 16. Xo = -1, X = 0, X2 = 1, x3 = 2 In Exercises 17-18, find lull and d(u, v) relative to the weighted Euclidean inner product (u, v) = 2u, , + 3uzu2 on R2. In Exercises 33-34, let u = (un, U2, u;) and v = (U1, Uz, U3). Show 17. u = (-3, 2) and v = (1, 7) that the expression does not define an inner product on R', and list all inner product axioms that fail to hold. 18. u = (-1,2) and v = (2, 5) 33. (u, v) = uju; + usvi+ ujob In Exercises 19-20, find Ipl| and d(p, q) relative to the standard inner product on P2. 34. (u, v) = u101 -4202+ Ugu3 19. p = -2+x+3x2, q=4-7x In Exercises 35-36, suppose that u and v are vectors in an inner prod- uct space. Rewrite the given expression in terms of (u, v), lull, and 20. p = -5+2x+ x, q=3+2x-4x2 v/12 In Exercises 21-22, find (| Ull and d(U, V) relative to the standard 35. (2v - 4u, u - 3v) 36. (5u + 6v, 4v - 3u) inner product on M22- 37. (Calculus required) Let the vector space P, have the inner 21. U = 3 -21 . v = [ 7 3] product 22. U = [_ 3 3]. = 16 8] (p. q) = / p(x)q(x) dx Find the following for p = 1 and q = x2. In Exercises 23-24, let a. (P, q) b. d(p. q) p= x+x and q=1+x2 c. P d. llall Find Ipl| and d(p, q) relative to the evaluation inner product on P3 at the stated sample points. 38. (Calculus required) Let the vector space P, have the inner product 23. XO = -2, X1 = -1, x2 = 0, x3 = 1 24. XO = -1, x1 = 0, X2 = 1, x3 = 2 (p, q) = p(x)q(x) dx In Exercises 25-26, find (lull and d(u, v) for the vectors u = (-1, 2) Find the following for p = 2x' and q = 1 -x3. and v = (2, 5) relative to the inner product on R2 generated by the a. (p, q) b. d(p, q) matrix A. c. Up d. ql 3 S 26. A = _1 31 (Calculus required) In Exercises 39-40, use the inner product In Exercises 27-28, suppose that u, v, and w are vectors in an inner product space such that (fig ) = [ f ( x8(x)dx (u, v) = 2, (v, w) = -6, (u, w) = -3 lull = 1, llvil = 2, 1/wil = 7 on C[0, 1] to compute (f, g). Evaluate the given expression. 39. f = cos 27tx, g = sin 27x 40. f = x, g= ex Working with Proofs 41. Prove parts (a) and (b) of Theorem 6.1.1. 12 10 42. Prove parts (c) and (d) of Theorem 6.1.1. 43. a. Letu = (uj, u2) andv = (v1, U2). Prove that the expression (u, v) = 3u, U, + 5u202 defines an inner product on R2 by showing that the inner product axioms hold. b. What conditions must k, and k2 satisfy for the expression (u, v) = Kjuju, + Kzuzu2 to define an inner product on R2? Justify your answer. 44. Prove that the following identity holds for vectors in any inner Wo product space. T1. (u, v) = 4//u + v/12 - Allu - v/12 45. Prove that the following identity holds for vectors in any inner product space. Ilu + v/12 + 1/u - v/12 = 21/ u/12 + 2/1v/12 46. The definition of a complex vector space was given in the first margin note in Section 4.1. The definition of a complex inner product on a complex vector space V is identical to that in Definition 1 except that scalars are allowed to be com- plex numbers, and Axiom 1 is replaced by (u, v) = (v, u). The remaining axioms are unchanged. A complex vector space with a complex inner product is called a complex inner prod- uct space. Prove that if V is a complex inner product space, then (u, kv) = k(u, v). 47. Prove that Formula (5) defines an inner product on Rn. T 48. a. Prove that if v is a fixed vector in a real inner product space V, then the mapping T : V - R defined by T(x) = (x, v) is a linear transformation. b. Let V = R3 have the