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Slides27-2010

Slides27-2010 - Vector Spaces and Eigenvalues Lecture XXVII...

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Vector Spaces and Eigenvalues: Lecture XXVII Charles B. Moss November 12, 2010 Charles B. Moss () Vector Spaces and Eigenvalues November 12, 2010 1 / 22

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1 Orthonormal Bases and Projections 2 Projection Matrices Idempotent Matrices 3 Eigenvalues and Eigenvectors 4 Kronecker Products Charles B. Moss () Vector Spaces and Eigenvalues November 12, 2010 2 / 22
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. 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) Charles B. Moss () Vector Spaces and Eigenvalues November 12, 2010 3 / 22

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which produces a set of orthogonal vectors, and then z i = y i q y 0 i y i (2) Example, the vectors x 1 = 1 3 4 , x 2 = 9 7 16 (3) span a plane in three dimension space. Setting y 1 = x 1 , y 2 is derived as y 2 = 9 7 16 - ( 9 7 16 ) 1 3 4 ( 1 3 4 ) 1 3 4 1 3 4 = 70 / 13 - 50 / 13 20 / 13 (4) Charles B. Moss () Vector Spaces and Eigenvalues November 12, 2010 4 / 22
The vectors can then be normalized to one. However, to test for orthogonality ( 1 3 4 ) 70 / 13 - 50 / 13 20 / 13 = 0 (5) 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 x = u + v (6) where u S and v is a vector that is orthogonal to every vector in S . Charles B. Moss () Vector Spaces and Eigenvalues November 12, 2010 5 / 22

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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 0 y = 0 , y S } .
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