Lecture27-2010 - Vector Spaces and Eigenvalues Charles B....

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Vector Spaces and Eigenvalues Charles B. Moss August 3, 2010 I. Orthonormal Bases and Projections A. 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 sim- plicity, we may want to form an orthogonal basis for this space. One way to form such a basis is the Gram-Schmit orthonormaliza- tion. In this procedure, we want to generate a new set of vectors { y 1 , y r } that are orthonormal. B. The Gram-Schmit process is y 1 = x 1 y 2 = x 2 x 0 2 y 1 y 0 1 y 1 y 1 y 3 = x 3 x 0 3 y 1 y 0 1 y 1 y 1 x 0 3 y 2 y 0 2 y 2 y 2 (1) which produces a set of orthogonal vectors, and then z i = y i q y 0 i y i (2) C. Example, the vectors x 1 = 1 3 4 ,x 2 = 9 7 16 (3) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVII Fall 2010 span a plane in three dimension space. Setting y 1 = x 1 , y 2 is derived as y 2 = 9 7 16 971 6 ± 1 3 4 134 ± 1 3 4 1 3 4 = 70 / 13 50 / 13 20 / 13 (4) The vectors can then be normalized to one. However, to test for orthogonality ± 70 / 13 50 / 13 20 / 13 =0 (5) D. Theorem 2.13 Every r -dimensional vector space, except the zero- dimensional space { 0 } , has an orthonormal basis. E. Theorem 2.14 Let { z 1 , ··· z r } be an orthornomal basis for some vector space S ,o f R m .T h e ne a c h x R m can be expressed uniquely as x = u + v (6) where u S and v is a vector that is orthogonal to every vector in S . F. Defnition 2.10 Let S be a vector subspace of R m h eo r - thogonal complement of S , denoted S , is the collection of all vectors in R m that are orthogonal to every vector in S :Th a ti s , S =( x : x R m and x 0 y , y S } .
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Lecture27-2010 - Vector Spaces and Eigenvalues Charles B....

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