This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture Note 16 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff ChakFu WONG 1 O RTHONORMAL B ASES IN R n ORTHONORMAL BASES IN R n 2 From our work with the natural bases • for R 2 , R 3 , and, in general, • for R n , we know that when these bases are present, the computations are kept to a minimum. ORTHONORMAL BASES IN R n 3 A subspace W of R n need not contain any of the natural basis, but in this lecture, we want to show that it has a basis with the same properties. That is, we want to show that W contains a basis S such that • every vector in S is of unit length and • every two vectors in S are orthogonal. The method used to obtain such a basis is the GramSchmidt process , which is presented in this lecture. ORTHONORMAL BASES IN R n 4 DEFINITION  • A set S = { u 1 , u 2 ,..., u k } in R n is called orthogonal if every pair of distinct vectors in S are orthogonal, that is, if u i · u j = 0 for i 6 = j . • An orthonormal set of vectors is an orthogonal set of unit vectors . That is, S = { u 1 , u 2 ,..., u k } is orthonormal if u i · u j = 0 for i 6 = j and u i · u i = 1 for i = 1 , 2 ,...,k. Normalizing refers to the process of dividing each vector in an orthogonal set S by its length so S is transformed into an orthonormal set . ORTHONORMAL BASES IN R n 5 Example 1 If x 1 = (1 , , 2) , x 2 = ( 2 , , 1) and x 3 = (0 , 1 , 0) , then { x 1 , x 2 , x 3 } is an orthogonal set in R 3 . The vectors u 1 = ( 1 √ 5 , , 2 √ 5 ) and u 2 = ( 2 √ 5 , , 1 √ 5 ) are unit vectors in the directions of x 1 and x 2 , respectively. • Since x 3 is also of unit length , it follows that { u 1 , u 2 , u 3 } is an orthonormal set . • Also, span { x 1 , x 2 , x 3 } is the same as span { u 1 , u 2 , u 3 } . ORTHONORMAL BASES IN R n 6 Example 2 The natural basis { (1 , , 0) , (0 , 1 , 0) , (0 , , 1) } is an orthonormal set in R 3 . More generally, the natural basis in R n is an orthonormal set. ORTHONORMAL BASES IN R n 7 Theorem 0.1 Let S = { u 1 , u 2 ,..., u k } be an orthogonal set of nonzero vectors in R n . Then S is linearly independent. Proof Consider the equation c 1 u 1 + c 2 u 2 + ··· c k u k = . (1) Taking the dot product of both sides of (1) with u i , 1 ≤ i ≤ k , we have ( c 1 u 1 + c 2 u 2 + ··· c k u k ) · u i = · u i . (2) By properties (c) and (d) of Theorem 0.3, Properties of dot Product , the left side of (2) is c 1 ( u 1 · u i ) + c 2 ( u 2 · u i ) + ··· + c k ( u k · u i ) , and the right side is . Since u j · u i = 0 if i 6 = j , (2) becomes 0 = c i ( u i · u i ) = c i k u i k 2 . (3) By (a) of Theorem 0.3, Properties of dot Product, k u i k 6 = 0 , since u i 6 = ....
View Full
Document
 Spring '06
 Dr.JeffChakFuWONG
 Linear Algebra, Algebra

Click to edit the document details