6.1 Write on paper please. Questions: 1,3, 7
6. (u, v) = {u101 + 5u202 8. A= -1 3 In Exercises 9-10, compute the standard inner product on M22 of th In Exercises 7-8, use the inner product on R2 generated by the matrix A to find (u, v) for the vectors u = (0, -3) and v = (6, 2). In Exercises 5-6, find a matrix that generates the stated weighted 9. U= / 4 8/. V = [ ] ]] 5. (u, v) = 24101+ 31202 7. A = 2 -3 inner product on R2. given matrices. = (u, 3u) + (u, 4v) - (2v, 3u) - (2v, 4v) = 3(u, u) + 4(u, v) - 6(v, u) - 8(v, v) = 3/|ul|2 + 4(u, v) - 6(u, v) -8//v/12 c. (u + v, w) f. llu - kull = 3/ /u/12 - 2(u, v) - 8/ /v//2 (u - 2v, 3u + 4v) = (u, 3u + 4v) - (2v, 3u + 4v) 4 . A= 1 2 _ ] (u, v) = 2u1 01 + 3u202 (u, v) = zu101 + 5U202 and let u = (1, 1), v = (3, 2), w = (0, -1), and k = 3. Com- 2. Follow the directions of Exercise 1 using the weighted In Exercises 3-4, compute the quantities in parts (a)-(f) of Exercise 1 b. (kv, w) e. d(u, v) 1. Let R2 have the weighted Euclidean inner product using the inner product on R2 generated by A pute the stated quantities. Euclidean inner product Exercise Set 6.1 a. (u, v) d. llvil 3. A= [ 1 350 CHAPTER 6 Inner Product Spaces 10. U = _3 31. = 16 81 27. a. (2v - w, 3u +2 28. a. (u - v - 2w, 4 In Exercises 11-12, find the standard inner product on P, of the given polynomials. In Exercises 29-30, ske 11. p = -2+x+3x2, q=4-7x2 product. 12. p = -5+2x+ x2, q=3+2x-4x2 29. (u, v) = juju, + In Exercises 13-14, a weighted Euclidean inner product on R2 is In Exercises 31-32, fi given for the vectors u = (u1, u2) and v = (v1, U2). Find a matrix for which the "unit ci that generates it. figure. 13. (u, v) = 3u,V, + 5u202 14. (u, v) = 4u,U, + 6u202 31. In Exercises 15-16, a sequence of sample points is given. Use the eval- uation inner product on P3 at those sample points to find (p, q) for the polynomials p= x+x and q=1+x2 15. XO = -2, X1 = -1, X2 = 0, x3 = 1 FIGURE Ex-3 16. Xo = -1, X1 = 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, V, + 3u202 on R2. In Exercises 33-34, 17. u = (-3,2) and v = (1, 7) that the expression all inner product a 18. u = (-1,2) and v = (2, 5) 33. (u, v) = uivi In Exercises 19-20, find Ilpll and d(p, q) relative to the standard inner product on P2. 34. (u, v) = u101 19. p = -2+x+3x2, q=4-7x2 In Exercises 35-36 20. p = -5+2x+x2, q=3+2x-4x2 uct space. Rewrite In Exercises 21-22, find || Ull and d( U, V) relative to the standard 35. (2v - 4u, u - inner product on M 22. 37. (Calculus re product 22. U = _3 3]. = 16 8] Find the foll
