Unformatted text preview: , U is an isometry. The linear map ± is U ± 1 ı M A ı U , where M A means multiplication by A . By Lemma 4.4.5 , M A is an isometry, so ± is an isometry. Thus (e) H) (a). Similarly, we have (a) H) (b) H) (d) H) (f) H) (a). n Proposition 4.4.8. The determinant of an orthogonal matrix is ˙ 1 . Proof. Let A be an orthogonal matrix. Since det .A/ D det .A t / , we have 1 D det .E/ D det .A t A/ D det .A t / det .A/ D det .A/ 2 : n Remark 4.4.9. If ± is a linear transformation of R n and A and B are the matrices of ± with respect to two different bases of R n , then det .A/ D det .B/ , because A and B are related by a similarity, A D VBV ± 1 , where V is a change of basis matrix. Therefore, we can, without ambiguity, deﬁne the determinant of ± to be the determinant of the matrix of ± with respect to any basis....
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Orthonormal basis

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