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 / 3 2 / 3 2 / 3 , - 2 / 3 2 / 3- 1 / 3 . Solution: We have ( 1 3 , 2 3 , 2 3 ) · (- 2 3 , 2 3 ,- 1 3 ) =- 2 9 + 4 9 +- 2 9 = 0. Hence, they are orthogonal. Observe that k ( 1 3 , 2 3 , 2 3 ) k = q 2 9 + 4 9 + 4 9 = 1 and k (- 2 3 , 2 3 ,- 1 3 ) k = q 4 9 + 4 9 + 1 9 = 1. Hence, the set is orthonormal. b) 1 2 1 , - 1 1 , 1- 1 1 . Solution: We have (1 , 2 , 1) · (- 1 , , 1) =- 1 + 0 + 1 = 0, (1 , 2 , 1) · (1 ,- 1 , 1) = 1- 2 + 1 = 0, and (- 1 , , 1) · (1 ,- 1 , 1) = 1 + 0 + 1 = 0. So, the set is orthogonal. Observe that k (1 , 2 , 1) k = √ 1 + 4 + 1 = √ 6, k (- 1 , , 1) k = √ 1 + 0 + 1 = √ 2, and k (1 ,- 1 , 1) k = √ 1 + 1 + 1 = √ 3. Hence, an orthonormal set is { 1 √ 6 (1 , 2 , 1) , 1 √ 2 (- 1 , , 1) , 1 √ 3 (1 ,- 1 , 1) } ....
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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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A4_soln - Math 235 Assignment 4 Solutions 1. Determine...

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