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Chapter 5 Orthogonality 1 The scalar product in Rn The product xTy is called the scalar product of x and y. In particular, if x=(x1, …, xn)T and y=(y1, …,yn)T, then xTy=x1y1+x2y2+‥‥+xnyn The Scalar Product in R2 and R3 Definition Let x and y be vectors in either R2 or R3. The distance between x and y is defined to be the number ‖x-y‖. Example If x=(3, 4)T and y=(-1, 7)T, then the distance between x and y is given by ‖y-x‖= 5 Theorem 5.1.1 If x and y are two nonzero vectors in either R2 or R3 and θ is the angle between them, then (1) xTy=‖x‖‖y‖cosθ Corollary 5.1.2 ( Cauchy-Schwarz Inequality) If x and y are vectors in either R2 or R3 , then (2) ︱xTy︱≤‖x‖‖y‖ with equality holding if and only if one of the vectors is 0 or one vector is a multiple of the other. Definition The vector x and y in R2 (or R3) are said to be orthogonal if xTy=0. Example (a) The vector 0 is orthogonal to every vector in R2. 3 4 (b) The vectors 2 and 6 are orthogonal in R2. 2 (c) The vectors and 3 1 1 are orthogonal in R3. 1 1 Scalar and Vector Projections x θ u z=x-p y p=αu x cos x y cos y xT y y The scalar is called the scalar projection of x and y, and the vector p is called the vector projection of x and y. Scalar projection of x onto y: T x y y Vector projection of x onto y: 1 xT y p u y T y y y y Example The point Q is the point on the line y 1 x that is 3 closet to the point (1, 4). Determine the coordinates of Q. (1, 4) y v w Q 1 x 3 Orthogonality in Rn The vectors x and y are said to be orthogonal if xTy=0. 2 Orthogonal Subspaces Definition Two subspaces X and Y of Rn are said to be orthogonal if xTy=0 for every x∈X and every y∈Y. If X and Y are orthogonal, we write X⊥Y. Example Let X be the subspace of R3 spanned by e1, and let Y be the subspace spanned by e2. Example Let X be the subspace of R3 spanned by e1 and e2, and let Y be the subspace spanned by e3. Definition Let Y be a subspace of Rn . The set of all vectors in Rn that are orthogonal to every vector in Y will be denoted Y⊥. Thus Y⊥={ x∈Rn︱xTy=0 for every y∈Y } The set Y⊥ is called the orthogonal complement of Y. Remarks 1. If X and Y are orthogonal subspaces of Rn, then X∩Y={0}. 2. If Y is a subspace of Rn, then Y⊥ is also a subspace of Rn. Fundamental Subspaces Theorem 5.2.1 ( Fundamental Subspaces Theorem) If A is an m×n matrix, then N(A)=R(AT) ⊥ and N(AT)=R(A) ⊥. Theorem 5.2.2 If S is a subspace of Rn, then dim S+dim S⊥=n. Furthermore, if {x1, …, xr} is a basis for S and {xr+1, …, xn} is a basis for S⊥, then {x1, …, xr, xr+1, …, xn} is a basis for Rn. Definition If U and V are subspaces of a vector space W and each w∈W can be written uniquely as a sum u+v, where u∈U and v∈V, then we say that W is a direct sum of U and V, and we write W=U V. Theorem 5.2.3 If S is a subspace of Rn, then Rn=S S⊥. Theorem 5.2.4 If S is a subspace of Rn, then (S⊥) ⊥=S. Theorem 5.2.5 If A is an m×n matrix and b∈Rm, then either there is a vector x∈Rn such that Ax=b or there is a vector y∈Rm such that ATy=0 and yTb≠0. Example Let 1 1 2 A 0 1 1 1 3 4 Find the bases for N(A), R(AT), N(AT), and R(A). 4 Inner Product Spaces Definition An inner product on a vector space V is an operation on V that assigns to each pair of vectors x and y in V a real number <x, y> satisfying the following conditions: Ⅰ. <x, x>≥0 with equality if and only if x=0. Ⅱ. <x, y>=<y, x> for all x and y in V. Ⅲ. <αx+βy, z>=α<x, z>+β<y, z> for all x, y, z in V and all scalars α and β. The Vector Space Rm×n Given A and B in Rm×n, we can define an inner product by m n A, B aijbij i 1 j 1 Basic Properties of Inner product Spaces If v is a vector in an inner product space V, the length or norm of v is given by v v, v Theorem 5.4.1 ( The Pythagorean Law ) If u and v are orthogonal vectors in an inner product space V, then uv u v 2 2 2 Example then If 1 1 A 1 2 3 3 A, B 6 A 5 B 6 and 1 1 B 3 0 3 4 Definition If u and v are vectors in an inner product space V and v≠0, then the scalar projection of u onto v is given by u, v v and the vector projection of u onto v is given by 1 u, v p v v v v, v Theorem 5.4.2 ( The Cauchy- Schwarz Inequality) If u and v are any two vectors in an inner product space V, then u, v u v Equality holds if and only if u and v are linearly dependent. 5 Orthonormal Sets Definition Let v1, v2, …, vn be nonzero vectors in an inner product space V. If <vi, vj>=0 whenever i≠j, then { v1, v2, …, vn} is said to be an orthogonal set of vectors. Example The set {(1, 1, 1)T, (2, 1, -3)T, (4, -5, 1)T} is an orthogonal set in R3. Theorem 5.5.1 If { v1, v2, …, vn} is an orthogonal set of nonzero vectors in an inner product space V, then v1, v2, …,vn are linearly independent. Definition An orthonormal set of vectors is an orthogonal set of unit vectors. The set {u1, u2, …, un} will be orthonormal if and only if u i , u j ij where 1 ij 0 if i j if i j Theorem 5.5.2 Let { u1, u2, …, un} be an orthonoemal basis for an inner product space V. If v n c u i 1 i i , then ci=<v, ui>. Corollary 5.5.3 Let { u1, u2, …, un} be an orthonoemal basis for an inner product space V. If u n a u i i 1 i and v n b u i 1 i i , then n u, v ai bi i 1 Corollary 5.5.4 If { u1, u2, …, un} is an orthonoemal basis for an inner product space V and v n c u i 1 n v ci2 2 i 1 i i , then Orthogonal Matrices Definition An n×n matrix Q is said to be an orthogonal matrix if the column vectors of Q form an orthonormal set in Rn. Theorem 5.5.5 An n×n matrix Q is orthogonal if and only if QTQ=I. Example cos For any fixed , the matrix Q sin is orthogonal. sin cos Properties of Orthogonal Matrices If Q is an n×n orthogonal matrix, then (a) The column vectors of Q form an orthonormal basis for Rn. (b) QTQ=I (c) QT=Q-1 (d) det(Q)=1 or -1 (e) The thanspose of an orthogonal matrix is an orthogonal matrix. (f) The product of two orthogonal matrices is also an orthogonal matrix. 6 The Gram-Schmidt Orthogonalization Process Theorem 5.6.1 ( The Gram-Schmidt Process) Let {x1, x2, …, xn} be a basis for the inner product space V. Let 1 x1 u1 x1 and define u2, …, un recursively by 1 u k 1 ( x k 1 p k ) x k 1 p k for k=1, …, n-1 where pk=<xk+1, u1>u1+<xk+1, u2>+‥‥<xk+1, uk>uk is the projection of xk+1 onto Span(u1, u2, …, uk). The set {u1, u2, …, un} is an orthonormal basis for V. Example Let 1 1 1 4 A 1 4 1 1 4 2 2 0 Find an orthonormal basis for the column space of A.