College Algebra Exam Review 222

College Algebra Exam Review 222 - ˇ ² is a unit vector...

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232 4. SYMMETRIES OF POLYHEDRA Corollary 4.4.10. The determinant of a linear isometry is ˙ 1 . Since det .AB/ D det .A/ det .B/ , det W O . n ; R / ! f 1 ; 1 g is a group homomorphism. Definition 4.4.11. The set of orthogonal n –by– n matrices with determi- nant equal to 1 is called the special orthogonal group and denoted SO . n ; R / . Evidently, the special orthogonal group SO . n ; R / is a normal sub- group of the orthogonal group of index 2, since it is the kernel of det W O . n ; R / ! f 1 ; 1 g . We next restrict our attention to three-dimensional space and explore the role of rotations and orthogonal reflections in the group of linear isome- tries. Recall from Section 4.3 that for any unit vector ˛ in R 3 , the plane P ˛ is f x W h x ; ˛ i D 0 g . The orthogonal reflection in P ˛ is the linear map j ˛ W x 7! x 2 h x ; ˛ i ˛ . The orthogonal reflection j ˛ fixes P ˛ pointwise and sends ˛ to ˛ . Let J ˛ denote the matrix of j ˛ with respect to the standard basis of R 3 . Call a matrix of the form J ˛ a reflection matrix . Let’s next sort out the role of reflections and rotations in SO . 2 ; R / . Consider an orthogonal matrix
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Unformatted text preview: ˇ ² is a unit vector and ± ± ı ² D ˙ ± ± ˇ ˛ ² : The vector ± ˛ ˇ ² can be written as ± cos .²/ sin .²/ ² for some angle ² , so the orthogonal matrix has the form ± cos .²/ ± sin .²/ sin .²/ cos .²/ ² or ± cos .²/ sin .²/ sin .²/ ± cos .²/ ² . The matrix R ± D ± cos .²/ ± sin .²/ sin .²/ cos .²/ ² is the matrix of the rotation through an angle ² , and has determinant equal to 1 . The matrix ± cos .²/ sin .²/ sin .²/ ± cos .²/ ² equals R ± J , where J D ± 1 ± 1 ² is the reflection matrix J D J O e 2 . The determinant of R ± J is equal to ± 1 . Now consider the situation in three dimensions. Any real 3–by–3 ma-trix has a real eigenvalue, since the characteristic polynomial is cubic with real coefficients. A real eigenvalue of an orthogonal matrix must be ˙ 1 because the matrix implements an isometry. Lemma 4.4.12. Any element of SO . 3 ; R / has C 1 as an eigenvalue....
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