This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full Document
 Spring '13
 MRR
 Math, Calculus, Dot Product, WI, AU, Det

Click to edit the document details