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Lecture 27-2007 - Vector Spaces and Eigenvalues Lecture...

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1 Vector Spaces and Eigenvalues Lecture XXVII I. Orthonormal Bases and Projections A. Suppose that a set of vectors 1 , r x x for a basis for some space S in m R 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 1 , r y y that are orthonormal. B. The Gram-Schmit process is: 1 1 2 1 2 2 1 1 1 3 1 3 2 3 3 1 1 2 2 ' ' ' ' ' ' y x x y y x y y y x y x y y x y y y y which produces a set of orthogonal vectors, and then 1 2 ' i i i i y z y y that produces a orthonormal vectors (vectors of length one). C. Example, from last lecture, we know that 16 7 9 , 4 3 1 2 1 x x spans a plane in three dimensional space. Setting 1 1 y x , 2 y is derived as: 2 1 70 9 7 16 3 13 9 1 4 50 7 3 13 1 16 4 20 1 3 4 3 13 4 y
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XXVII Professor Charles Moss Fall 2005 2 The vectors can then be normalized to one. However, to test for orthogonality: 70 13 50 1 3 4 0 13 20 13 D. Theorem 2.13 Every r -dimensional vector space, except the zero- dimensional space 0 , has an orthonormal basis. E. Theorem 2.14 Let 1 , r z z be an orthornomal basis for some vector space S , of m R . Then each m x R can be expressed uniquely as x u v where u S and v is a vector that is orthogonal to every vector in S . F. Definition 2.10 Let S be a vector subspace of m R . The orthogonal complement of S , denoted S , is the collection of all vectors in m R that are orthogonal to every vector in S : That is, : 0 m S x x R and x y y S .
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