College Algebra Exam Review 224

College Algebra Exam Review 224 - 2 . (d) Show that any...

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234 4. SYMMETRIES OF POLYHEDRA Exercises 4.4 4.4.1. (a) Show that j ˛ is isometric. (b) Show that det .j ˛ / D ± 1 . (c) If ± is a linear isometry, show that ±j ˛ ± ± 1 D j ±.˛/ . (d) If A is any orthogonal matrix, show that AJ ˛ A ± 1 D J . (e) Conclude that the matrix of j ˛ with respect to any orthonormal basis is a reflection matrix. 4.4.2. Show that the set of isometries of R n is a group. Show that the set of isometries ± satisfying ±. 0 / D 0 is a subgroup. 4.4.3. Show that the following are equivalent for a matrix A : (a) A is orthogonal. (b) The columns of A are an orthonormal basis. (c) The rows of A are an orthonormal basis. 4.4.4. (a) Show that the matrix R J D R ² JR ± ² is the matrix of the reflection in the line spanned by ± cos ² sin ² ² : Write J ² D R J . (b) The reflection matrices are precisely the elements of O . 2 ; R / with determinant equal to ± 1 . (c) Compute J ² 1 J
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Unformatted text preview: 2 . (d) Show that any rotation matrix R is a product of two reection matrices. 4.4.5. Show that an element of SO . 3 ; R / is a product of two reection matrices. A matrix of a rotationreection is a product of three reec-tion matrices. Thus any element of O . 3 ; R / is a product of at most three reection matrices. 4.5. The Full Symmetry Group and Chirality All the geometric gures in three-dimensional space that we have con-sidered the polygonal tiles, bricks, and the regular polyhedra admit reection symmetries. For reasonable gures, every symmetry is imple-mented by by a linear isometry of R 3 ; see Proposition 1.4.1 . The rotation...
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