Chapter 4 General Vector Spaces Face Recognition ( á¹¢å ± 彐嬀)

4.6 Orthonormal Vectors and ProjectionsLA 04_45Definition: A set of vectors is said to be an orthogonal () set if every pair () of vectors in the set is orthogonal.• The set is said to be an orthonormal () set if itis orthogonal and each vector is a unit () vector.• Example 1(1,0,0),0,35,45,0,45,35(1) orthogonal: 3 4 4 3 ( 1,0,0) 0, , 0;(1,0,0) 0, , 0; 0, 5 5 5 5 n n ( n 1)nn 2 2Check . Now 3 3 times, 4 6(2) unit vector22 2 2 0 1, 0, 02 ( 1,0,0) 1 01, 0,02 24535452353545354535,45 0,45,3520;1Definition:Orthogonal basis: basis and orthogonal ( + )Orthonormal basis: basis and orthonormal ( + , || || = 1)LA 04_47Standard Bases• R 2 : {(1, 0), (0, 1)}• R 3 : {(1, 0, 0), (0, 1, 0), (0, 0, 1)} orthonormal bases• R n : {(1, …, 0), …, (0, …, 1)}Theorem 4.20: Let {u 1 , …, u n } be an orthonormal basis and v beany vector. Thenv ( v u1) u1 ( v u2) u2 ( v u n) u nProof: v = c 1 u 1 + … + c n u n uniquely v • u i = (c 1 u 1 + … + c n u n ) • u i = c i u i • u i = c i• Do not need Gauss-Jordan elimination as shown in slide 17.ATheorem 4.19An orthogonal set of nonzero vectors is linearly independent.LA 04_46Proof Let {v 1 , …, v m } be an orthogonal set of nonzero vectorsc 1 v 1 + c 2 v 2 + … + c m v m = 0 c i = 0 ?c1v1(vc1v1c2cv22v2vcmcvmmv)mvv00v• If j i v j v i = 0 (mutually () orthogonal)• c i v i v i = 0• Since v i is a nonzero vector v i v i 0 c i = 0.• Letting i = 1, …, m, we get c 1 = 0, …, c m = 0.Orthogonal Matrices (1/2)LA 04_48 Definition: An orthogonal matrix is an invertible matrixthat has the property A 1 = A t . That is, A A t =A t A = I Theorem 4.21: Equivalent statements— (a) A is orthogonal— (b) The column vectors of A form an orthonormal set.— (c) The row vectors of A form an orthonormal set.— Proof: We will prove (a) (b) only.Let A = [A 1 … A n ]. By assumption,A I AA1nA A 1n I AA1nAA11AA1nAAnn 1 0 0 1 ,,tttiittiitti