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# Solving these equations gives t = 2 x • ˆ w so

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Unformatted text preview: Solving these equations gives t =- 2 x • ˆ w so that Sx = x- 2( x • ˆ w ) ˆ w. 22 Chapter 7 This matrix S is called “reflection in the direction ˆ w ,” and we denote it S ( ˆ w ). PROBLEM 7–26. Prove that the matrix S = S ( ˆ w ) satisfies a. S ˆ w =- ˆ w , b. S 2 = I , c. S ∈ O ( n ), d. det S =- 1. PROBLEM 7–27. Prove that S ˆ w = I- 2( w i w j ) . Now we return to the situation of a matrix A ∈ O ( n ) with det A =- 1. We know from Problem 7–22 that there is a unit vector ˆ w satisfying A ˆ w =- ˆ w . Now just define B = S ( ˆ w ) A . Clearly, B ∈ SO( n ). Thus, A = S ( ˆ w ) B. Thus A is the product of a reflection S ( ˆ w ) in a direction ˆ w and a matrix B ∈ SO( n ) with B ˆ w = ˆ w . PROBLEM 7–28. It made no difference whether we chose to write B = SA or B = AS , for SA = AS . Prove this by noting that SA ˆ w = AS ˆ w and then considering those vectors x which are orthogonal to ˆ w . In particular, for n = 3 the matrix A has the representation A = S ( ˆ w ) R ( ˆ w,θ ) = R ( ˆ w,θ ) S ( ˆ w ) . PROBLEM 7–29. Show that in this n = 3 situation A = (cos θ ) I- (1 + cos θ )( w i w j ) + sin θ - w 3 w 2 w 3- w 1- w 2 w 1 . Cross product 23 PROBLEM 7–30. Show that in case n = 2 every matrix A ∈ O (2) with det A =- 1 is equal to a reflection matrix S ( ˆ w ). PROBLEM 7–31. Refer back to inversion in the unit sphere in R n , p. 6–34: f ( x ) = x k x k 2 . Prove that Df ( x ) = k x k- 2 S x k x k ¶ . Thus we say that inversion in the unit sphere is an orientation- reversing conformal mapping. PROBLEM 7–32. Let u , v be linearly independent vectors in R 3 and let u , v also be vectors in R 3 . a. Suppose that A ∈ SO(3) satisfies Au = u , Av = v . Prove that u and v are linearly independent and k u k = k u k , k v k = k v k , u • v = u • v . ( * ) b. Conversely, suppose that ( * ) is satisfied. Prove that u and v are linearly indepen- dent, and that there exists a unique A ∈ SO(3) such that Au = u , Av = v . (HINT: A ( u v u × v ) = ( u v u × v ).) 24 Chapter 7 PROBLEM 7–33. Under the assumptions of the preceding problem prove that ( u v u × v )- 1 = k u × v k- 2 ( v × ( u × v )- u × ( u × v ) u × v ) t ....
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Solving these equations gives t = 2 x • ˆ w so that Sx =...

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