This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ./ sin ./ cos ./ 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 reection, or a reectionrotation, that is, the product of a reection 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 reection. Then itself is a reection. 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 reection j v and a rotation about the line spanned by v . n...
View
Full
Document
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

Click to edit the document details