Introduction to Inner Product Spaces in Linear Algebra

1. Lecture 10 Vector & Inner Product Space Last Time - Applications of Vector Space - Coordinates and Change of Basis Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_11_2007

2. Lecture 10: Inner Product Spaces Today • Coordinates and Change of Basis (Cont.) • Length and Dot Product in Rn • Inner Product Spaces Reading Assignment: Secs 4.7,4.8, 5.1,5.2 Next Time • Orthonormal Bases:Gram-Schmidt Process • Mathematical Models and Least Square Analysis • Applications Reading Assignment: Secs 5.3-5.5

3. What Have You Actually Learned about Change of Basis So Far? Ex. Superposition Principle in Circuit Analysis

4. Today • Coordinates and Change of Basis (Cont.) • Length and Dot Product in Rn • Inner Product Spaces

6. Notes: • Thm 4.20: (The inverse of a transition matrix) If P is the transition matrix from a basis B'to a basis B in Rn, then (1) P is invertible (2) The transition matrix from B to B' is P–1

7. Thm 4.21: (Transition matrix from B to B') Let B={v1, v2, … , vn} and B'={u1, u2,… , un} be two bases for Rn. Then the transition matrix P–1 from B to B' can be found by using Gauss-Jordan elimination on the n×2n matrix as follows.

10. Ex 5: (Finding a transition matrix) B={(–3, 2), (4,–2)} and B' ={(–1, 2), (2,–2)} are two bases for R2 (a) Find the transition matrix from B' to B. (b) (c) Find the transition matrix from B to B' .

11. G.J.E. Sol: (a) BB' IP-1 [I2 : P] = (the transition matrix from B' to B) (b)

12. G.J.E. • Check: (c) B'B IP-1 (the transition matrix from B toB')

13. Rotation of the Coordinate Axes

15. Ex 6: (Coordinate representation in P3(x)) Find the coordinate matrix of p = 3x3-2x2+4 relative to the standard basis in P3(x), S = {1, 1+x, 1+ x2, 1+ x3}. Sol: p = 3(1) + 0(1+x) + (–2)(1+x2 ) + 3(1+x3) [p]s =

16. Ex: (Coordinate representation in M2x2) Find the coordinate matrix of x = relative to the standardbasis in M2x2. B = Sol:

17. Keywords in Section 4.7: • coordinates of x relative to B：x相對於B的座標 • coordinate matrix：座標矩陣 • coordinate vector：座標向量 • change of basis problem：基底變換問題 • transition matrix from B' to B：從 B'到 B的轉移矩陣

18. Today • Coordinates and Change of Basis • Length and Dot Product in Rn • Inner Product Spaces

19. Notes: Properties of length is called a unit vector. 5.1 Length and Dot Product in Rn • Length: The length of a vector in Rn is given by • Notes: The length of a vector is also called its norm.

22. Ex 1: (a)In R5, the length of is given by (b)In R3 the length of is given by (vis a unit vector)

23. Ex: the standard unit vectorin R2: the standard unit vectorin R3: • A standard unit vectorin Rn: • Notes: (Two nonzero vectors are parallel) u and v have the same direction u and v have the opposite direction

24. Pf: • Thm 5.1: (Length of a scalar multiple) Let v be a vector in Rn and c be a scalar. Then

25. Pf: v is nonzero (u has the same direction as v) (u has length 1 ) • Thm 5.2: (Unit vector in the direction of v) If v is a nonzero vector in Rn, then the vector has length 1 and has the same direction as v. This vector u is called the unit vector in the direction of v.

26. Notes: (1) The vector is called the unit vector in the direction of v. (2) The process of finding the unit vector in the direction of v is called normalizing the vector v.

27. Sol: is a unit vector. • Ex 2: (Finding a unit vector) Find the unit vector in the direction of , and verify that this vector has length 1.

28. Notes: (Properties of distance) (1) (2) if and only if (3) • Distance between two vectors: The distance between two vectors u and v in Rn is

29. Ex 3: (Finding the distance between two vectors) The distance between u=(0, 2, 2) and v=(2, 0, 1) is

30. Ex 4: (Finding the dot product of two vectors) The dot product of u=(1, 2, 0, -3) and v=(3, -2, 4, 2) is • Dot productin Rn: The dot product of and is the scalar quantity

31. Thm 5.3: (Properties of the dot product) If u, v, and w are vectors in Rn and c is a scalar, then the following properties are true. (1) (2) (3) (4) (5), and if and only if

32. Euclidean n-space: Rn was defined to be the set of all order n-tuples of real numbers. When Rn is combined with the standard operations of vector addition, scalar multiplication, vector length, and the dot product, the resulting vector space is called Euclidean n-space.

33. Ex 5: (Finding dot products) Sol: • (b) (c) (d) • (e)

34. Ex 6: (Using the properties of the dot product) Given Find Sol:

35. Thm 5.4: (The Cauchy - Schwarz inequality) If u and v are vectors in Rn, then ( denotes the absolute value of ) • Ex 7: (An example of the Cauchy - Schwarz inequality) Verify the Cauchy - Schwarz inequality for u=(1, -1, 3) and v=(2, 0, -1) Sol:

36. The angle between two vectors in Rn: Same direction Opposite direction • Note: The angle between the zero vector and another vector is not defined.

37. Ex 8: (Finding the angle between two vectors) Sol: u and v have opposite directions.

38. Orthogonal vectors: Two vectors u and v in Rn are orthogonal if • Note: The vector 0 is said to be orthogonal to every vector.

39. Ex 10: (Finding orthogonal vectors) Determine all vectors in Rn that are orthogonal to u=(4, 2). Sol: Let

40. Thm 5.5: (The triangle inequality) If u and v are vectors in Rn, then Pf: • Note: Equality occurs in the triangle inequality if and only if the vectors u and v have the same direction.

41. Thm 5.6: (The Pythagorean theorem) If u and v are vectors in Rn, then u and v are orthogonal if and only if

42. Dot product and matrix multiplication: (A vector in Rn is represented as an n×1 column matrix)

43. Keywords in Section 5.1: • length: 長度 • norm: 範數 • unit vector: 單位向量 • standard unit vector : 標準單位向量 • normalizing: 單範化 • distance: 距離 • dot product: 點積 • Euclidean n-space: 歐基里德n維空間 • Cauchy-Schwarz inequality: 科西-舒瓦茲不等式 • angle: 夾角 • triangle inequality: 三角不等式 • Pythagorean theorem: 畢氏定理

44. Today • Applications of Vector Space • Coordinates and Change of Basis • Length and Dot Product in Rn • Inner Product Spaces

45. 5.2 Inner Product Spaces • Inner product: Let u, v, and w be vectors in a vector space V, and let c be any 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 axioms. (1) (2) (3) (4)and if and only if

46. Note: A vector space V with an inner product is called an inner product space. Vector space: Inner product space: • Note:

47. Ex 1:(The Euclidean inner product for Rn) Show that the dot product in Rn satisfies the four axioms of an inner product. Sol: By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.

48. Ex 2:(A different inner product for Rn) Show that the function defines an inner product on R2, where and . Sol:

49. Note: (An inner product on Rn)

50. Ex 3: (A function that is not an inner product) Show that the following function is not an inner product on R3. Sol: Let Axiom 4 is not satisfied. Thus this function is not an inner product on R3.