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HW6_solutions

# HW6_solutions - Math 167 homework 6 solutions 3.4.2 Project...

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Math 167 homework 6 solutions November 11, 2010 3.4.2: Project b = 0 3 0 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

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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 bracketrightbigg bracketleftbigg 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 > 0 . Since q 2 = b ( q T 1 b ) q 1 bardbl b ( q T 1 b ) q 1 bardbl , the second input vector b must be of the form b = λ 2 q 2 + λ 2 q 1 , where λ 2 , λ 3 R and λ 2 > 0 3.4.14 From the nonorthogonal a, b, c find orthonormal vectors q 1 , q 2 , q 3 : a = 1 1 0 , b = 1 0 1 , c =
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HW6_solutions - Math 167 homework 6 solutions 3.4.2 Project...

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