inner_product_spaces0

inner_product_spaces0 - 1 INNER PRODUCT SPACES Definition...

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Unformatted text preview: 1 INNER PRODUCT SPACES Definition Two vectors u and v in an inner product space are called orthogonal if an inner product is equal to zero: < u , v > = 0. A Geometric Link between Nullspace and Row Space of a Matrix Theorem 6.2.6 If A is an m × n matrix, then (a) The nullspace of A and the row space of A are or- thogonal complements in R n with respect to the Euclidean inner product. (b) The nullspace of A T and the column space of A are orthogonal complements in R m with respect to the Eu- clidean inner product. 2 EXAMPLE Basis for an Orthogonal Comple- ment Let W be the subspace of R 5 spanned by the vectors w 1 = (2, 2, -1, 0, 1), w 2 = (-1, -1, 2, -3, 1), w 3 = (1, 1,-2, 0,-1), w 4 = (0,0, 1, 1,1) Find a basis for the orthogonal complement of W . Solution The space W spanned by w 1 ,w 2 ,w 3 , and w 4 is the same as the row space of the matrix A= 2 2- 1 0 1- 1- 1 2- 3 1 1 1- 2 0- 1 1 1 1...
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This note was uploaded on 06/20/2008 for the course AM 025b taught by Professor Kiruscheva during the Winter '07 term at UWO.

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inner_product_spaces0 - 1 INNER PRODUCT SPACES Definition...

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