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Unformatted text preview: orthogonal if A t is the inverse of A . Exercise 4.4.3 gives another characterization of orthogonal matrices. Lemma 4.4.5. If A is an orthogonal matrix, then the linear map x 7! A x is an isometry. Proof. For all x 2 R n , jj A x jj 2 D h A x ;A x i D h A t A x ; x i D h x ; x i D jj x jj 2 . n Lemma 4.4.6. Let F D f f i g be an orthonormal basis. A set f v i g is an orthonormal basis if and only if the matrix A D Œa ij Ł D Œ h v i ; f j i Ł is an orthogonal matrix. Proof. The i th row of A is the coefﬁcient vector of v i with respect to the orthonormal basis F . According to Remark 4.4.3 , the inner product of...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Matrices

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