Section 5.1 Length and Dot Product in ℝ n

1. Section 5.1Length and Dot Product in ℝn

2. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. The dot product of v and w is v ∙ w = __________________________________

3. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. The length of v (magnitude of v, norm of v) is || v || = ___________________________________ The length of v may also be computed using the formula _________________________________________

4. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. The distance between v and w is d(v, w) = _____________________________

5. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. The vectors v and w are orthogonal if ________________________________________

6. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. A unit vector in the direction of v is given by ___________________ A unit vector in the direction opposite of v is given by __________________

7. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. A vector in the direction of v with magnitude of c is given by ____________

8. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. The angle between vectors v and w is given by ____________________________________

9. Let v = ‹v1­­, v2, v3, . . . , vn›and w = ‹w1­­, w2, w3, . . . , wn›be vectors in ℝn. || v + w ||2 = || v ||2 + || w ||2 if __________________________________

10. Ex. Let v = ‹4, 2›. (a) Determine all vectors which are orthogonal to v.

11. Ex. Let v = ‹4, 2›. (b) Find a vector parallel to v, but with a magnitude five times that of v.

12. Ex. Let v = ‹4, 2›. (c) Find a vector in the opposite direction of v, but with a magnitude of five.

13. Properties of the dot product. Let u, v, and w be vectors in ℝn and let c be a scalar. 1. u ∙ v = v ∙ u 2. u ∙ (v + w) = u ∙ v + u ∙ w 3. c (u ∙ v) = (cu) ∙ v = u ∙ (cv) 4. v ∙ v ≥ 0 and v ∙ v = 0 if and only if v =0.

14. Section 5.2Inner Product Spaces

15. Let u, v, and w be vectors in a vector space V, and let c be a scalar. An inner product on V is a function that associates a real number <u,v> with each pair of vectors u and v and satisfies the following: 1. < u , v > = < v , u > 2. < u , v + w > = < u , v > + < u , w > 3. c < u , v > = < cu , v > = <u , cv > 4. < v , v > ≥ 0 and < v , v > = 0 if and only if v= 0.

16. Ex. An inner product on M2,2 :

17. Ex. An inner product on M2,2 :

18. Ex. An inner product on P2 .

19. Ex. An inner product on C[a,b]:

20. Def. Let v and w be vectors in an inner product space V. (a) The magnitude (norm) of v is || v || = ___________________________ (b) The distance between vand w is d(v, w) = _______________________ (c) The angle between v andw is found by the formula ________________________________________ (d)v and w are orthogonal (v ⊥w) if _____________________________

21. Note: || v + w ||2 = || v ||2 + || w ||2 if v and w are orthogonal.

22. Ex. Let f (x) = x , g(x) = x2 , and h(x) = x2 + 1 be functions in the inner product space C[0, 1]. (a) Compute || f ||

23. Ex. Let f (x) = x , g(x) = x2 , and h(x) = x2 + 1 be functions in the inner product space C[0, 1]. (b) Compute d( f, g) and d (g , h)

24. Ex. Let f (x) = x , g(x) = x2 , and h(x) = x2 + 1 be functions in the inner product space C[0, 1]. (c) Compare the distance between f and g with the distance between g and h.

25. Recall the projection vector of v onto w in ℝn: projwv =

26. The projection vector of v onto w in an inner product space V:

27. Ex. Let f (x) = x and g(x) = x2 be functions in the inner product space C[a, b]. Find the projection of f onto g.

28. Theorem: Let v and w be two vectors in an inner product space V with w ≠ 0. Then d(v, projw v) ≤ d(v, cw), with equality only when c =

29. Section 5.3Orthonormal Bases & the Gram-Schmidt Process

30. Def. A set of vectors S is orthogonal if every pair of vectors in S is orthogonal. If in addition, every vector in S is a unit vector then S is orthonormal.

31. Examples: (i) In ℝ3 the set of basis vectors {i, j, k} form an orthonormal set.

32. Examples: (ii) In ℝ2 the set { (2, 2), (−3, 3) } is an orthogonal set of vectors but not orthonormal. We can turn it into an orthonormal set though.

33. Examples: (iii) Is the set {x+1, x−1, x2} an orthogonal set in P2? Is it an orthonormal set in P2? (Use the standard inner product in P2 )

34. Examples: (iv) Is the set {x+1, x−1, x2} an orthogonal set in C[0,1]? Is it an orthonormal set in C[0,1]? (Use the standard inner product in C[0,1] )

35. Ex. Create an orthonormal basis for ℝ3 that includes a vector in the direction of (3,0,3).

36. Ex. Verify that the set { 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), . . . . . , sin(nx), cos(nx) } is orthogonal in C[0, 2π] and then turn it into an orthonormal basis.

37. Ex. Verify that the set { 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), . . . . . , sin(nx), cos(nx) } is orthogonal in C[0, 2π] and then turn it into an orthonormal basis.

38. Ex. Verify that the set { 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), . . . . . , sin(nx), cos(nx) } is orthogonal in C[0, 2π] and then turn it into an orthonormal basis.

39. Ex. Verify that the set { 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), . . . . . , sin(nx), cos(nx) } is orthogonal in C[0, 2π] and then turn it into an orthonormal basis.

40. Ex. Verify that the set { 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), . . . . . , sin(nx), cos(nx) } is orthogonal in C[0, 2π] and then turn it into an orthonormal basis.

41. Def. The coordinate matrix of a vector w with respect to a basis B= { v1, v2, v3, . . . . , vn } is the column matrix [c1 , c2 , c3 , . . . . , cn]T , if w can be expressed as a linear combination of basis vectors with the coordinates c1 , c2 , c3 , . . . . , cn (eg if w = c1v1 + c2v2 + c3v3 + . . . . + cnvn )

42. Ex. (a) Find the coordinate matrix of (2, 3, 5) in ℝ3 with respect to the standard basis {i, j, k} and the standard inner product.

43. Ex. (b) Find the coordinate matrix of (2, 3, 5) in ℝ3 with respect to the standard basis {k, j, i} and the standard inner product.

44. Ex. (c) Find the coordinate matrix of (2, 3, 5) in ℝ3 with respect to the basis { (1,1,1) , (1,2,3) , (−1,0,4) } and the standard inner product.

45. Theorem: Let B = { v1, v2, v3, . . . . , vn } be an orthonormal basis. The coordinates of w= c1v1 + c2v2 + c3v3 + . . . . + cnvn can be computed by ck= < w , vk >

46. Ex. Give the coordinate matrix of (5, −5, 2) with respect to the orthonormal basis { (3⁄5, 4⁄5, 0), (−4⁄5, 3⁄5, 0), (0,0,1) }.

47. Gram-Schmidt Orthonormalization Process: Let B = { v1, v2, v3, . . . . , vn } be a basis for an inner product space. First form B′ = { w1, w2, w3, . . . . , wn } where the wk are given by

48. Gram-Schmidt Orthonormalization Process: Let B = { v1, v2, v3, . . . . , vn } be a basis for an inner product space. First form B′ = { w1, w2, w3, . . . . , wn } where the wk are given by w1 = v1 ⁞ Then form B″ = { u1, u2, u3, . . . . , un } where each uk is given by

49. Ex. Use the Gram-Schmidt process on the basis { (1,1), (0,1) } to find an orthonormal basis for ℝ2.

50. Ex. Use the Gram-Schmidt process on { (1,1,0), (1,2,0), (0,1,2) } to find an orthonormal basis for ℝ3.