College Algebra Exam Review 223

College Algebra Exam Review 223 - ./ sin ./ cos ./ 3 5 ; so...

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4.4. LINEAR ISOMETRIES 233 Proof. Let A 2 SO . 3 ; R / , let ± be the linear isometry x 7! A x , and let v be an eigenvector with eigenvalue ˙ 1 . If the eigenvalue is C 1 , there is nothing to do. So suppose the eigenvalue is ± 1 . The plane P D P v orthogonal to v is invariant under A , because if x 2 P , then h v ;A x i D ±h A v ;A x i D ±h v ; x i D 0 . The restriction of ± to P is also orthogonal, and since 1 D det .±/ D . ± 1/. det j P / , ± j P must be a reflection. But a reflection has an eigenvalue of C 1 , so in any case A has an eigenvalue of C 1 . n Proposition 4.4.13. An element A 2 O . 3 ; R / has determinant 1 if and only if A implements a rotation. Proof. Suppose A 2 SO . 3 ; R / . Let ± denote the corresponding linear isometry x 7! A x . By the lemma, A has an eigenvector v with eigen- value 1 . The plane P orthogonal to v is invariant under ± , and det j P / D det .±/ D 1 , so ± j P is a rotation of P . Hence, ± is a rotation about the line spanned by v . On the other hand, if ± is a rotation, then the matrix of ± with respect to an appropriate orthonormal basis has the form 2 4 1 0 0 0 cos .²/ ± sin
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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 reection-rotation, 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 orthogo-nal 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...
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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.

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