1 / 56

Lecture 10 Inner Product Spaces

Lecture 10 Inner Product Spaces. Last Time Rank of a Matrix and Systems of Linear Equations (Cont.) Coordinates and Change of Basis Applications of Vector Space. Elementary Linear Algebra R. Larsen et al. (6th Edition) TKUEE 翁慶昌 -NTUEE SCC_12_2008. Lecture 10: Inner Product Spaces.

denverl
Download Presentation

Lecture 10 Inner Product Spaces

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 10 Inner Product Spaces Last Time Rank of a Matrix and Systems of Linear Equations (Cont.) Coordinates and Change of Basis Applications of Vector Space Elementary Linear Algebra R. Larsen et al. (6th Edition) TKUEE翁慶昌-NTUEE SCC_12_2008

  2. Lecture 10:Inner Product Spaces Today Change of Basis (Cont.) Length and Dot Product in Rn Inner Product Spaces Orthonormal Bases:Gram-Schmidt Process Reading Assignment: Secs 5.1-5.3

  3. Application to Optimization

  4. 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.

  5. 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' .

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

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

  8. Rotation of the Coordinate Axes

  9. 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 =

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

  11. 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

  12. Lecture 10:Inner Product Spaces Today Change of Basis (Cont.) Length and Dot Product in Rn Inner Product Spaces Orthonormal Bases:Gram-Schmidt Process

  13. 5.1 Length and Dot Product in Rn Length: The length of a vector in Rn is given by • Notes: Properties of length is called a unit vector. Q: Why? • Notes: The length of a vector is also called its norm. Q: Length in non-orthogonal basis?

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

  15. A standard unit vectorin Rn: • Ex: the standard unit vectorin R 2: the standard unit vectorin R 3: • Notes: (Two nonzero vectors are parallel) u and v have the same direction u and v have the opposite direction

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

  17. 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. Pf: v is nonzero (u has the same direction as v) (u has length 1 )

  18. 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.

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

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

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

  22. Dot productin Rn: The dot product of and is the scalar quantity • 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

  23. 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

  24. 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.

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

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

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

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

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

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

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

  32. 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.

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

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

  35. 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

  36. Lecture 10:Inner Product Spaces Today Change of Basis (Cont.) Length and Dot Product inRn Inner Product Spaces Orthonormal Bases:Gram-Schmidt Process

  37. 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 10 - 39

  38. Note: • Note: A vector space V with an inner product is called an inner product space. Vector space: Inner product space: 10 - 40

  39. 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. 10 - 41

  40. Ex 2:(A different inner product for Rn) Show that the function defines an inner product on R2, where and . Sol: 10 - 42

  41. Note: (An inner product on Rn) 10 - 43

  42. Ex 3: (A function that is not an inner product) Show that the following function is not an inner product on R 3. Sol: Let Axiom 4 is not satisfied. Thus this function is not an inner product on R3. 10 - 44

  43. Thm 5.7: (Properties of inner products) Let u,v, and w be vectors in an inner product space V, and let c be any real number. (1) (2) (3) • Norm (length) of u: • Note: 10- 45

  44. Distance between u and v: • Angle between two nonzero vectors u and v: • Orthogonal: u and v are orthogonal if . 10 - 46

  45. Notes: (1) If , then v is called a unit vector. (2) (the unit vector in the direction of v) not a unit vector 5 - 47

  46. Sol: • Ex 6: (Finding inner product) is an inner product 10 - 48

  47. Properties of norm: (1) (2) if and only if (3) • Properties of distance: (1) (2) if and only if (3) 10 - 49

  48. Thm 5.8： Let u and v be vectors in an inner product spaceV. (1) Cauchy-Schwarz inequality: (2) Triangle inequality: (3) Pythagorean theorem : u and v are orthogonal if and only if Theorem 5.4 Theorem5.5 Theorem5.6 10 - 50

More Related