Lecture 27-2007 - Lecture XXVII Orthonormal Bases and...

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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 Gram-Schmit 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 r-dimensional vector space, except the zero-dimensional 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 } ....
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Lecture 27-2007 - Lecture XXVII Orthonormal Bases and...

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