Report

Chapter 5 Inner Product Spaces n 5.1 Length and Dot Product in R Notes: The length of a vector is also called its norm. Notes: 1 v 0 2 v 1 v 3 v 0 iff is called a unit vector. v 0 5-1 5-2 5-3 • Notes: The process of finding the unit vector in the direction of v is called normalizing the vector v. • A standard unit vector in Rn: e 1 , e 2 , , e n 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 5-4 Notes: (Properties of distance) (1) d (u , v ) 0 (2) d (u , v ) 0 (3) d (u , v ) d ( v , u ) if and only if u v 5-5 5-6 • 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. 5-7 Dot product and matrix multiplication: u1 u2 u u n v1 v2 v v n u v u v [u 1 T u2 (A vector u ( u1 , u 2 , , u n ) in Rn is represented as an n×1 column matrix) v1 v2 [u v u v u v ] un ] 1 1 2 2 n n v n 5-8 Note: The angle between the zero vector and another vector is not defined. 5-9 Note: The vector 0 is said to be orthogonal to every vector. 5-10 Note: Equality occurs in the triangle inequality if and only if the vectors u and v have the same direction. 5-11 5.2 Inner Product Spaces • Note: u v dot product ( Euclidean n inner product for R ) u , v general inner product for vector space V 5-12 Note: A vector space V with an inner product is called an inner product space. Vector space: V , , Inner product space: V , , , , 5-13 5-14 Note: || u || 2 〈 u , u 〉 5-15 Properties of norm: (1) || u || 0 (2) || u || 0 if and only if u0 (3) || c u || | c | || u || 5-16 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 ) 5-17 Note: If v is a init vector, then 〈 v , v 〉 || v || 2 1. The formula for the orthogonal projection of u onto v takes the following simpler form. proj v u u , v v 5-18 5-19 5.3 Orthonormal Bases: Gram-Schmidt Process S v 1 , v 2 , , v n V S v 1 , v 2 , , v n V vi, v j 0 1 vi, v j 0 i j i j Note: If S is a basis, then it is called an orthogonal basis or an orthonormal basis. 5-20 5-21 5-22 5-23 5-24 5-25 5.4 Mathematical Models and Least Squares Analysis 5-26 Orthogonal complement of W: Let W be a subspace of an inner product space V. (a) A vector u in V is said to orthogonal to W, if u is orthogonal to every vector in W. (b) The set of all vectors in V that are orthogonal to W is called the orthogonal complement of W. W {v V | v , w 0 , w W } W Notes: (1) 0 V (read “ W perp”) (2) V 0 5-27 • Notes: W is a subspace (1) W is a subspace (2) W W (3) of V of V 0 (W ) W Ex: If V R , W x axis 2 Then (1) W y - axis (2) W W (3) (W is a subspace of R 2 ( 0 , 0 ) ) W 5-28 5-29 5-30 5-31 • Notes: (1) Among all the scalar multiples of a vector u, the orthogonal projection of v onto u is the one that is closest to v. (2) Among all the vectors in the subspace W, the vector proj W v is the closest vector to v. 5-32 • The four fundamental subspaces of the matrix A: N(A): nullspace of A N(AT): nullspace of AT R(A): column space of A R(AT): column space of AT 5-33 5-34 Least squares problem: Ax b m n n 1 m 1 (A system of linear equations) (1) When the system is consistent, we can use the Gaussian elimination with back-substitution to solve for x (2) When the system is inconsistent, how to find the “best possible” solution of the system. That is, the value of x for which the difference between Ax and b is small. Least squares solution: Given a system Ax = b of m linear equations in n unknowns, the least squares problem is to find a vector x in Rn that minimizes Ax b with respect to the Euclidean inner product on Rn. Such a vector is called a least squares solution of Ax = b. 5-35 A M x R mn n A x CS ( A ) ( CS A is a subspace m of R ) W CS ( A ) Let A xˆ proj W b ( b A xˆ ) CS ( A ) b A xˆ ( CS ( A )) NS ( A ) A ( b A xˆ ) 0 i.e. A A xˆ A b (the normal equations of the least squares problem Ax = b) 5-36 • Note: The problem of finding the least squares solution of A x b is equal to he problem of finding an exact solution of the associated normal system A A xˆ A b . Thm: For any linear system A x b , the associated normal system A A xˆ A b is consistent, and all solutions of the normal system are least squares solution of Ax = b. Moreover, if W is the column space of A, and x is any least squares solution of Ax = b, then the orthogonal projection of b on W is proj W b A x 5-37 • Thm: If A is an m×n matrix with linearly independent column vectors, then for every m×1 matrix b, the linear system Ax = b has a unique least squares solution. This solution is given by x ( A A) 1 A b Moreover, if W is the column space of A, then the orthogonal projection of b on W is proj W b A x A ( A A ) 1 A b 5-38 5.5 Applications of Inner Product Spaces 5-39 5-40 • Note: C[a, b] is the inner product space of all continuous functions on [a, b]. 5-41 5-42