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Unformatted text preview: ORTHOGONAL SPACES JOS E MALAG ONL OPEZ Now we will get different descriptions of R n . The first case is in terms of orthogonal subspaces. Orthogonal Complement Definition 1. Let V be a subspace of R n . The orthogonal comple ment of V , denoted as V , is the set of all the vectors in R n that are orthogonal to all the vectors in V . In other words, for any vectorv in V , and any vectorw in V , we have vectorv vectorw = 0. Remark 2 . Notice that if { vectorv 1 , . . . ,vectorv r } is a basis for V , then vectorw is in V if and only if vectorw vectorv i = 0, for 1 i r . Theorem 3. V is a subspace of R n . Proof . Let { vectorv 1 , . . . ,vectorv r } be a basis for V . Let vectorw 1 , vectorw 2 be any two vectors in V , and let and be any two scalars. Then vectorv i ( vectorw 1 + vectorw 2 ) = ( vectorv i vectorw 1 ) + ( vectorv i vectorw 2 ) = 0 + 0 = 0 , for any 1 i r . Q.E.D. Example 4. If A is any m n matrix, and V = Row( A ), we have by definition that V = (Row( A )) = Null( A ). Remark 5 . From the definition of the transpose of a matrix we know that Col( A T ) = Row( A ). Thus, we have that Col ( A T ) = Row( A ) = (Null( A )) , which is equivalent to having Col ( A ) = Row ( A T )...
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 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra

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