College Algebra Exam Review 224

College Algebra Exam Review 224 - ² 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 2 . (d) Show that any rotation matrix R ² is a product of two reflection matrices. 4.4.5. Show that an element of SO . 3 ; R / is a product of two reflection matrices. A matrix of a rotation–reflection is a product of three reflec-tion matrices. Thus any element of O . 3 ; R / is a product of at most three reflection matrices. 4.5. The Full Symmetry Group and Chirality All the geometric figures in three-dimensional space that we have con-sidered — the polygonal tiles, bricks, and the regular polyhedra — admit reflection symmetries. For “reasonable” figures, every symmetry is imple-mented by by a linear isometry of R 3 ; see Proposition 1.4.1 . The rotation...
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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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