A9_soln - Math 235 Assignment 9 Solutions 1 Let u =(2 3i 2 i and v =(4 i 4 i Use the standard inner product on Cn to calculate < u v > < v u > and

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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 k ~v k . 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 k ~v k = p < ~v,~v > = q (4 - i, 4 + i ) · (4 - i, 4 + i ) = p (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 ² . 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
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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A9_soln - Math 235 Assignment 9 Solutions 1 Let u =(2 3i 2 i and v =(4 i 4 i Use the standard inner product on Cn to calculate < u v > < v u > and

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