HW6_solutions

HW6_solutions - Math 167 homework 6 solutions November 11,...

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Unformatted text preview: Math 167 homework 6 solutions November 11, 2010 3.4.2: Project b = 3 onto each of the orthonormal vectors a 1 = 1 3 2 2 − 1 and a 2 = 1 3 − 1 2 2 , and then find its projection p onto the plane of a 1 and a 2 . Since a 1 and a 2 are orthonormal vectors, the matrix P of the projection onto the subspace spanned by a 1 and a 2 is given by P = QQ T , where Q = bracketleftbig a 1 a 2 bracketrightbig = 1 3 2 − 1 2 2 − 1 2 . If follows that p = QQ T b = Q bracketleftbigg 2 2 bracketrightbigg = 1 3 2 8 2 . 3.4.4 If Q 1 and Q 2 are orthogonal matrices, so that Q T Q = I , show that Q 1 Q 2 is also orthogonal. If Q 1 is rotation through θ , and Q 2 is rotation through φ , what is Q 1 Q 2 ? Can you find the trigonometric identities for sin( θ + φ ) and cos( θ + φ ) in the matrix multiplication Q 1 Q 2 ? 1 Since ( Q 1 Q 2 )- 1 = Q- 1 2 Q- 1 1 = Q T 2 Q T 1 = ( Q 1 Q 2 ) T , the matrix Q 1 Q 2 is orthog- onal. If Q 1 is the rotation through θ and Q 2 is the rotation through φ , then Q 1 Q 2 is a rotation through θ + φ : Q 1 Q 2 = bracketleftbigg cos θ − sin θ sin θ cos θ bracketrightbiggbracketleftbigg cos φ − sin φ sin φ cos φ bracketrightbigg = bracketleftbigg cos( θ + φ ) − sin( θ + φ ) sin( θ + φ ) cos( θ + φ ) bracketrightbigg . By carrying out the actual matrix multiplication Q 1 Q 2 and comparing it with the matrix on the right-hand side of the last equation, it follows that cos θ cos φ − sin θ sin φ = cos( θ + φ ) and cos θ sin φ + sin θ cos φ = sin( θ + φ ) . 3.4.10 If q 1 and q 2 are the outputs from Gram-Schmidt, what were the possible input vectors a and b ? Since q 1 = a bardbl a bardbl , the first input vector a must be of the form a = λ 1 q 1 , where λ 1 ∈ R and λ 1 > ....
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This note was uploaded on 03/18/2012 for the course MAT MAT 167 taught by Professor Rolandw.freund during the Fall '10 term at UC Davis.

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HW6_solutions - Math 167 homework 6 solutions November 11,...

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