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Unformatted text preview: Math 235 Assignment 9 Solutions 1. Let ~u = (1 + i, 2 i ) and ~v = (3 i, 3 + i ). Use the standard inner product on C n to calculate < ~u,~v > , < ~v,~u > , and k ~v k . Solution: We have < ~u,~v > = (1 + i, 2 i ) · (3 i, 3 + i ) = (1 + i )(3 + i ) + (2 i )(3 i ) = 2 + 4 i + 5 5 i = 7 i < ~v,~u > = (3 i, 3 + i ) · (1 + i, 2 i ) = (3 i )(1 i ) + (3 + i )(2 + i ) = 2 4 i + 5 + 5 i = 7 + i k ~v k = p < ~v,~v > = q (3 i, 3 + i ) · (3 i, 3 + i ) = p (3 i )(3 + i ) + (3 + i )(3 i ) = √ 20 2. Determine which of the following matrices is unitary. a) A = (1 + i ) / √ 7 5 / √ 35 (1 + 2 i ) / √ 7 (3 + i ) / √ 35 . Solution: Observe that 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...
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This note was uploaded on 04/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math

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