Chapter 6.1, Write on paper please Questions : 11, 15, 21

350 CHAPTER 6 Inner Product Spaces 10. U = _3 3). = 18 8) 27. a. (2v - w, 3u + 2w) b. lu + v!! b. 112w - vil In Exercises 11-12, find the standard inner product on Pz of the given 28. a. (u - v - 2w, 4u + v) polynomials. In Exercises 29-30, sketch the unit circle in R using the given inner 11. p = -2 + x+ 3x2, q=4-7x2 product. 29. (u, v) = tu,,+ 18202 30. ( u, v ) = 24101 + 120/2 12. p = -5+ 2x + x2, q=3+2x-4x2 In Exercises 13-14, a weighted Euclidean inner product on R2 is In Exercises 31-32, find a weighted Euclidean inner product on R2 for which the "unit circle" is the ellipse shown in the accompanying given for the vectors u = (u, u2) and v = (v1, v2). Find a matrix that generates it. figure. 13. (u, v) = 31, v, + 5u202 14. (u, v) = 46,0, + 64212 32. 31. 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=xtx and q=1+x2 15. X0 = -2, X1 = -1, X2 = 0, X3 = 1 FIGURE Ex-32 FIGURE Ex-31 16. X0 = -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) = 24,v, + 3uzu2 on R2. In Exercises 33-34, let u = (u] , U2, u;) and v = (U1, U2, V3). Show 17. u = (-3, 2) and v = (1, 7) that the expression does not define an inner product on R3, and list all inner product axioms that fail to hold. 18. u = (-1, 2) and v = (2, 5) 33. (u, v) = ujvi + uhuh + ujus In Exercises 19-20, find Ipl| and d(p, q) relative to the standard inner product on P2- 34. (u, v) = uv1 - uzu2 + Ugly 19. p = -2 +x+ 3x?, q=4-7x2 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 In Exercises 21-22, find || Ul| and d(U, V) relative to the standard 35. (2v - 4u, u - 3v) 36. (5u + 6v, 4v - 3u) inner product on M22. 21. U = [3 -2], v =[ ]] 37. (Calculus required) Let the vector space P2 have the inner product =1-3 31. W = 18 81 (p. q) = p(x)q(xx) dx Find the following for p = 1 and q = x2. In Exercises 23-24, let a.