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 XXVII Orthonormal Bases and Projections Suppose that a set of vectors { x 1 ,, x r } for a basis for some space S in R m space such that r m . For mathematical simplicity, we may want to form an orthogonal basis for this space. One way to form such a basis is the GramSchmit orthonormalization. In this procedure, we want to generate a new set of vectors { y 1 , y r } that are orthonormal. 2 2 2 3 1 1 1 3 3 3 1 1 1 1 2 2 2 1 1 ' ' ' ' ' ' y y y x y y y x x y y y y y x x y x y = = = ( 29 2 1 ' i i i i y y y z = = = 16 7 9 , 4 3 1 2 1 x x ( 29 ( 29  =  = 13 20 13 50 13 70 4 3 1 4 3 1 4 3 1 4 3 1 16 7 9 16 7 9 2 y The vectors can then be normalized to one. However, to test for orthogonality: ( 29 13 20 13 50 13 70 4 3 1 =  Theorem 2.13 Every rdimensional vector space, except the zerodimensional space { 0} , has an orthonormal basis. Theorem 2.14 Let { z 1 , z r } be an orthornomal basis for some vector space S , of R m . Then each x R m can be expressed uniquely as were u S and v is a vector that is orthogonal to every vector in S . v u x + = Definition 2.10 Let S be a vector subspace of R m . The orthogonal complement of S , denoted S , is the collection of all vectors in R m that are orthogonal to every vector in S : That is, S ={ x : x R m and x y =0 for all y S } ....
View Full
Document
 Fall '09
 CARRIKER

Click to edit the document details