College Algebra Exam Review 220

College Algebra Exam Review 220 - orthogonal if A t is the...

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230 4. SYMMETRIES OF POLYHEDRA related by h A x ; y i D h x ;A t y i for all n –by– n matrices A , and all x ; y 2 R n . Definition 4.4.1. An isometry of R n is a map ± W R n ±! R n that pre- serves distance, d.±. a /;±. b // D d. a ; b / , for all a ; b 2 R n . You are asked to show in the Exercises that the set of isometries is a group, and the set of isometries ± satisfying ±. 0 / D 0 is a subgroup. Lemma 4.4.2. Let ± be an isometry of R n such that ±. 0 / D 0 . Then h ±. a /;±. b / i D h a ; b i for all a ; b 2 R n . Proof. Since ±. 0 / D 0 , ± preserves norm as well as distance, jj ±. x / jj D jj x jj for all x . But since h a ; b i D .1=2/. jj a jj 2 C jj b jj 2 ± d. a ; b / 2 / , it follows that ± also preserves inner products. n Remark 4.4.3. Recall the following property of orthonormal bases of R n : If F D f f i g is an orthonormal basis, then the expansion of a vector x with respect to F is x D P i h x ; f i i f i . If x D P i x i f i and y D P i y i f i , then h x ; y i D P i x i y i . Definition 4.4.4. A matrix A is said to be
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Unformatted text preview: orthogonal if A t is the inverse of A . Exercise 4.4.3 gives another characterization of orthogonal matrices. Lemma 4.4.5. If A is an orthogonal matrix, then the linear map x 7! A x is an isometry. Proof. For all x 2 R n , jj A x jj 2 D h A x ;A x i D h A t A x ; x i D h x ; x i D jj x jj 2 . n Lemma 4.4.6. Let F D f f i g be an orthonormal basis. A set f v i g is an orthonormal basis if and only if the matrix A D Œa ij Ł D Œ h v i ; f j i Ł is an orthogonal matrix. Proof. The i th row of A is the coefficient vector of v i with respect to the orthonormal basis F . According to Remark 4.4.3 , the inner product of...
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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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