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midterm09sols

midterm09sols - L Vandenberghe EE103 Midterm Solutions...

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L. Vandenberghe 10/27/09 EE103 Midterm Solutions Problem 1 (10 points). The cross-product a × x of two 3-vectors a = ( a 1 , a 2 , a 3 ) and x = ( x 1 , x 2 , x 3 ) is defined as the vector a × x = a 2 x 3 a 3 x 2 a 3 x 1 a 1 x 3 a 1 x 2 a 2 x 1 . 1. Assume a is fixed and nonzero. Show that the function f ( x ) = a × x is a linear function of x , by giving a matrix A that satisfies f ( x ) = Ax for all x . 2. Is the matrix A you found in part 1 singular or nonsingular? 3. Verify that A T A = ( a T a ) I aa T . 4. Use the observations in parts 1 and 3 to show that for nonzero x , bardbl a × x bardbl = bardbl a bardbl bardbl x bardbl | sin θ | where θ is the angle between a and x . Solution. 1. a × x = Ax with A = 0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 . 2. Singular, because Aa = 0 so A has a nonzero vector in its nullspace. 3. Working out the matrix product gives A T A = a 2 2 + a 2 3 a 1 a 2 a 1 a 3 a 1 a 2 a 2 1 + a 2 3 a 2 a 3 a 1 a 3 a 2 a 3 a 2 1 + a 2 3 = a 2 1 + a 2 2 + a 2 3 0 0 0 a 2 1 + a 2 2 + a 2 3 0 0 0 a 2 1 + a 2 2 + a 2 3 a 2 1 a 1 a 2 a 1 a 3 a 1 a 2 a 2 2 a 2 a 3 a 1 a 3 a 2 a 3 a 2 3 = ( a T a ) I aa T . 1
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4. From the expression in part 3, bardbl a × x bardbl 2 = x T A T Ax = x T (( a T a ) I aa T ) x = ( a T a )( x T x ) ( x T a ) 2 = bardbl a bardbl 2 bardbl x bardbl 2 ( bardbl a bardblbardbl x bardbl cos θ ) 2 = ( bardbl a bardblbardbl x bardbl sin θ ) 2 . 2
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Problem 2 (15 points). Let A be defined as A = I + BB T where B is a given orthogonal n × m matrix (not necessarily square).
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