A4_soln - Math 235 Assignment 4 Solutions 1. Determine...

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Unformatted text preview: Math 235 Assignment 4 Solutions 1. Determine which sets of vectors are orthonormal in R 3 under the standard inner product. If a set is only orthogonal, normalize the vectors to produce an orthonormal set. a) 1 / 2 2 / 3 , - 4 3 . Solution: We have ( 1 2 , 2 3 , 0) (- 4 , 3 , 0) =- 2 + 2 + 0 = 0. Hence, they are orthogonal. We have k ( 1 2 , 2 3 , 0) k = q 1 4 + 4 9 + 0 = q 25 36 = 5 6 and k (- 4 , 3 , 0) k = 16 + 9 + 0 = 5. Hence, an orthonormal set is 3 / 5 4 / 5 , - 4 / 5 3 / 5 b) 1 / 3 1 / 3 1 / 3 , 1 / 6 1 / 6- 2 / 6 , 1 / 2 1 / 2 . Solution: Observe that ( 1 2 , 1 2 , 0) ( 1 6 , 1 6 ,- 2 6 ) = 1 3 , hence the set is not even orthogonal. 2. Determine which sets of vectors are orthonormal in M (2 , 2) under the inner product < A, B > = tr( A T B ). If a set is only orthogonal, normalize the vectors to produce an)....
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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.

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A4_soln - Math 235 Assignment 4 Solutions 1. Determine...

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