Unformatted text preview: .²/ sin .²/ cos .²/ 3 5 ; so ± has determinant 1. n Proposition 4.4.14. An element of O . 3 ; R / n SO . 3 ; R / implements either an orthogonal reﬂection, or a reﬂectionrotation, that is, the product of a reﬂection j ˛ and a rotation about the line spanned by ˛ . Proof. Suppose A 2 O . 3 ; R / n SO . 3 ; R / . Let ± denote the corresponding linear isometry x 7! A x . Let v be an eigenvector of A with eigenvalue ˙ 1 . If the eigenvalue is 1 , then the restriction of ± to the plane P orthogonal to v has determinant ± 1 , so is a reﬂection. Then ± itself is a reﬂection. If the eigenvalue is ± 1 , then the restriction of ± to P has determinant 1 , so is a rotation. In this case ± is the product of the reﬂection j v and a rotation about the line spanned by v . n...
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, det.

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