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Math 222 - Selected Homework Solutions from Chapter 5.5, 5.6. Instructor - Al Boggess Fall 1998 Section 5 10. Prove that the transpose of an orthogonal matrix is orthogonal. If Q is orthogonal, then Qt Q = I . Therefore Qt is the inverse of Q. To show that Qt is orthogonal, we must show that (Qt )t Qt = I Now (Qt )t = Q. So equation is equivalent to showing QQt =I However, since Qt is the inverse of Q, clearly QQt = I as well. 11. The acute angle between two vectors x and y can be determined by the Law of Cosines: cos = jjxxjjjjyyjj Likewise, let be the acute angle between Qx and Qy. It is determined by Qx Qy cos = jjQx jj jjQyjj Since Q is orthogonal, Qx Qy is the same as x y and jjQxjj = jjxjj; jjQyjj = jjyjj. Thus cos = cos , which implies = . 25. a) Let f = 1 and g = 2x , 1. Then hf; gi = Z 1 0 f g dx = Z 1 0 1 (2x , 1) dx = x2 , xj10 = 0 So, f and g are orthogonal. b) Z 1 jj1jj2 = 12 dx = 1 0 So jj1jj = 1. We also have jjgjj = 2 Z 1 0 g dx = 2 Z 1 (2x , 1)2 dx = (1=6)(2x , 1)3 j10 = 1=3 0 p So jj2x , 1jj = 1= 3. p p c) Let f1 = f =jjf jj = 1 and f2 = g=jjgjj = 3(1 , 2x). Let h = x. Then the best least squares approximation of h by an element in S = the span of f1 and f2 is H = hh; f1 if1 + hh; f2 if2 We now compute the inner products: hh; f1 i = and Thus Z 1 0 0 p px dx = (2=3)x3=2 j1 = 2=3 hh; f2 i = 3 Z 1 0 px(1 , 2x) dx = ,2p3 15 p 2 3 f = (2=3) , (2=5)(1 , 2x) H = (2=3) , 15 2 Section 6 11. If v3 is in the span of v1 ; v2 , then the Gram Schmidt orthogonalization process will fail to produce three orthonormal vectors which span v1 ; v2 ; v3 . Note that this space is at most two dimensional (since v1 , v2 and v3 are linearly dependent). However, three orthonormal vectors must be linearly independent and therefore the span of these vectors would be three dimensional. The step that fails in the Gram Schmidt process is that if p is the projection of v3 onto the space spanned by v1 , v2 , then p = v3 (since v3 is contained in this space). Therefore v3 , p = 0 and one cannot divide by its length to get the third orthonormal vector. 2