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A9_soln - Math 235 Assignment 9 Solutions 1 Let u =(2 3i 2...

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Math 235 Assignment 9 Solutions 1. Let u = ( - 2 - 3 i, 2 + i ) and v = (4 - i, 4 + i ). Use the standard inner product on C n to calculate < u, v > , < v, u > , and v . Solution: We have < u, v > = ( - 2 - 3 i, 2 + i ) · (4 - i, 4 + i ) = ( - 2 - 3 i )(4 + i ) + (2 + i )(4 - i ) = - 5 - 14 i + 9 + 2 i = 4 - 12 i < v, u > = (4 - i, 4 + i ) · ( - 2 - 3 i, 2 + i ) = (4 - i )( - 2 + 3 i ) + (4 + i )(2 - i ) = - 5 + 14 i + 9 - 2 i = 4 + 12 i v = < v, v > = (4 - i, 4 + i ) · (4 - i, 4 + i ) = (4 - i )(4 + i ) + (4 + i )(4 - i ) = 34 2. Determine which of the following matrices is unitary. a) A = (1 + i ) / 7 - 5 / 35 (1 + 2 i ) / 7 (3 + i ) / 35 . Solution: Let u = ((1 + i ) / 7 , (1 + 2 i ) / 7), and v = ( - 5 / 35 , (3 + i ) / 35). Then we have < u, u > = 1 7 [(1 + i )(1 - i ) + (1 + 2 i )(1 - 2 i )] = 1 7 [2 + 5] = 1 < v, v > = 1 35 [( - 5)( - 5) + (3 + i )(3 - i )] = 1 35 [25 + 10] = 1 < u, v > = 1 7 35 [(1 + i )( - 5) + (1 + 2 i )(3 - i )] = 1 7 35 [ - 5 - 5 i + 5 + 5 i ] = 0 Hence, { u, v } is an orthonormal basis for C 2 and so A is unitary. b) A = (1 + i ) / 6 (1 + i ) / 3 2 i/ 6 i/ 3 . Solution: Let u = ((1 + i ) / 6 , 2 i/ 6), and v = (1 + i ) / 3 , i/ 3). Then < u, v > = 1 6 3 [(1 + i )(1 - i ) + 2 i ( - i )] = 4 18 . Hence u and v are not orthogonal so A is not unitary.
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2 3. a) Verify that u = (1 + i, 1 , 2) is orthogonal to v = (1 - i, 2 i, 0).
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