CHAPTER 5 INNER PRODUCT SPACES 5.1 Length and Dot Product in Rn 5.2 Inner Product Spaces 5.3 Orthonormal Bases: Gram-Schmidt Process 5.4 Mathematical Models and Least Square Analysis 5.5 Applications of Inner Product Space Elementary Linear Algebra 投影片設計製作者 R. Larson (8 Edition) 淡江大學 電機系 翁慶昌 教授 CH 5 Linear Algebra Applied Electric/Magnetic Flux (p.240) Heart Rhythm Analysis (p.255) Work (p.248) Revenue (p.266) Torque (p.277) 2/101 5.1 Length and Dot Product in Rn Length: n The length of a vector v ( v 1 , v 2 , , v n ) in R is given by 2 2 2 || v|| v1 v2 vn Notes: The length of a vector is also called its norm. Notes: Properties of length 1 v 0 2 v 1 v is called a unit vector. 3 v 0 iff v 0 4 cv c v Elementary Linear Algebra: Section 5.1, p.232 3/101 Ex 1: (a) In R5, the length of v ( 0 , 2 , 1 , 4 , 2 ) is given by ||v|| 02 (2)2 12 42 (2)2 25 5 (b) In R3 the length of v ( 2 , 2 , 3 ) is given by 17 17 17 2 2 2 2 2 3 17 || v|| 1 17 17 17 17 (v is a unit vector) Elementary Linear Algebra: Section 5.1, p.232 4/101 n A standard unit vector in R : e1,e2,,en 1,0,,0,0,1,,0,0,0,,1 Ex: the standard unit vector in R2: i, j 1,0,0,1 the standard unit vector in R3: i, j,k 1,0,0,0,1,0,0,0,1 Notes: (Two nonzero vectors are parallel) u cv 1 c 0 u and v have the same direction 2 c 0 u and v have the opposite direction Elementary Linear Algebra: Section 5.1, p.232 5/101 Thm 5.1: (Length of a scalar multiple) Let v be a vector in Rn and c be a scalar. Then ||cv|| |c||| v|| Pf: v (v1 , v2 ,, vn ) cv (cv1 , cv2 ,, cvn ) ||cv|| || (cv1 , cv2 , , cvn ) || 2 2 2 (cv1) (cv2 ) (cvn ) 2 2 2 2 c (v1 v2 vn ) 2 2 2 |c| v1 v2 vn |c| || v|| Elementary Linear Algebra: Section 5.1, p.233 6/101 Thm 5.2: (Unit vector in the direction of v) v If v is a nonzero vector in Rn, then the vector u ||v|| has length 1 and has the same direction as v. This vector u is called the unit vector in the direction of v. Pf: 1 v is nonzero v 0 0 v 1 u v (u has the same direction as v) v v 1 ||u|| || v|| 1 (u has length 1 ) || v|| || v|| Elementary Linear Algebra: Section 5.1, p.233 7/101 Notes: v (1) The vector is called the unit vector in the direction of v. ||v|| (2) The process of finding the unit vector in the direction of v is called normalizing the vector v. Elementary Linear Algebra: Section 5.1, p.233 8/101 Ex 2: (Finding a unit vector) Find the unit vector in the direction of , and verify that this vector has length 1. Sol: v (3, 1, 2) v 32 12 22 14 v (3 , 1, 2) 1 3 1 2 (3 , 1, 2) , , || v|| 32 (1)2 22 14 14 14 14 2 2 2 3 1 2 14 1 14 14 14 14 v v is a unit vector. Elementary Linear Algebra: Section 5.1, p.233 9/101 Distance between two vectors: The distance between two vectors u and v in Rn is d(u , v) ||u v|| Notes: (Properties of distance) (1) d(u , v) 0 (2) d (u , v ) 0 if and only if u v (3) d(u , v) d(v , u) Elementary Linear Algebra: Section 5.1, p.234 10/101 Ex 3: (Finding the distance between two vectors) The distance between u = (0, 2, 2) and v = (2, 0, 1) is d(u , v) ||u v|| || (0 2 , 2 0 , 2 1) || (2)2 22 12 3 Elementary Linear Algebra: Section 5.1, p.234 11/101 n Dot product in R : The dot product of u ( u 1 , u 2 , , u n ) and v (v1 , v2 ,, vn ) is the scalar quantity u v u1v1 u2v2 unvn 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 u v (1)(3) (2)(2) (0)(4) (3)(2) 7 Elementary Linear Algebra: Section 5.1, p.235 12/101 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) u v v u (2) u (v w) u v u w (3) c(u v) (cu) v u (cv) (4) v v ||v||2 (5) v v 0 , and v v 0 if and only if v 0 Elementary Linear Algebra: Section 5.1, p.235 13/101 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. Elementary Linear Algebra: Section 5.1, p.235 14/101 Ex 5: (Finding dot products) u (2 , 2) , v (5 , 8), w (4 , 3) (a)u v (b) ( u v ) w (c) u ( 2 v ) (d) || w || 2 (e) u (v 2w) Sol: (a) u v (2)(5) (2)(8) 6 (b) (u v)w 6w 6(4 , 3) (24 , 18) (c) u (2v) 2(u v) 2(6) 12 (d) || w ||2 w w (4)(4) (3)(3) 25 (e) v 2w (5 (8) , 8 6) (13 , 2) u (v 2w) (2)(13) (2)(2) 26 4 22 Elementary Linear Algebra: Section 5.1, p.236 15/101 Ex 6: (Using the properties of the dot product) Given u u 39 u v 3 v v 79 Find (u 2v) (3u v) Sol: (u 2v)(3u v) u (3u v) 2v (3u v) u (3u) u v (2v)(3u) (2v) v 3(u u) u v 6(v u) 2(v v) 3(u u) 7(u v) 2(v v) 3(39) 7(3) 2(79) 254 Elementary Linear Algebra: Section 5.1, p.236 16/101 Thm 5.4: (The Cauchy - Schwarz inequality) If u and v are vectors in Rn, then | u v | || u || || v || ( | u v | denotes the absolute value of u v ) 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: u v 1, uu 11, v v 5 u v 1 1 u v u u v v 11 5 55 u v u v Elementary Linear Algebra: Section 5.1, p.237 17/101 n The angle between two vectors in R : u v cos , 0 || u |||| v || Same Opposite u v 0 u v 0 u v 0 direction direction 0 0 2 2 2 cos 1 cos 0 cos 0 cos 0 cos 1 Note: The angle between the zero vector and another vector is not defined. Elementary Linear Algebra: Section 5.1, p.238 18/101 Ex 8: (Finding the angle between two vectors) u (4 , 0 , 2 , 2) v (2 , 0 , 1,1) Sol: u uu 42 02 22 22 24 v v v 22 02 12 12 6 u v (4)(2) (0)(0) (2)(1) (2)(1) 12 u v 12 12 cos 1 || u || || v|| 24 6 144 u and v have opposite directions. (u 2v) Elementary Linear Algebra: Section 5.1, p.238 19/101 Orthogonal vectors: Two vectors u and v in Rn are orthogonal if u v 0 Note: The vector 0 is said to be orthogonal to every vector. Elementary Linear Algebra: Section 5.1, p.238 20/101 Ex 10: (Finding orthogonal vectors) Determine all vectors in Rn that are orthogonal to u=(4, 2). Sol: u (4 , 2) Let v (v1 , v2 ) u v (4 , 2) (v1 , v2 ) 4v1 2v2 1 4 2 0 1 0 0 2 t v , v t 1 2 2 t v ,t, t R 2 Elementary Linear Algebra: Section 5.1, p.238 21/101 Thm 5.5: (The triangle inequality) If u and v are vectors in Rn, then ||u v|| ||u|| ||v|| Pf: ||u v||2 (u v) (u v) u (u v) v (u v) u u 2(u v) v v ||u||2 2(u v) || v||2 ||u||2 2 |u v| || v||2 ||u||2 2 ||u||||v|| ||v||2 (||u|| ||v||)2 ||u v|| ||u|| ||v|| Note: Equality occurs in the triangle inequality if and only if the vectors u and v have the same direction. Elementary Linear Algebra: Section 5.1, p.239 22/101 Thm 5.6: (The Pythagorean theorem) If u and v are vectors in Rn, then u and v are orthogonal if and only if ||u v||2 ||u||2 ||v||2 Elementary Linear Algebra: Section 5.1, p.239 23/101 Dot product and matrix multiplication: u1 v1 u v (A vector u ( u , u , , u ) in Rn u 2 v 2 1 2 n is represented as an n×1 column matrix) un vn v1 v u v uT v [u u u ] 2 [u v u v u v ] 1 2 n 1 1 2 2 n n vn Elementary Linear Algebra: Section 5.1, p.240 24/101 Key Learning in Section 5.1 ▪ Find the length of a vector and find a unit vector.