A5_soln - Math 235 Assignment 5 Solutions 1. Use the...

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Unformatted text preview: Math 235 Assignment 5 Solutions 1. Use the Gram-Schmidt procedure to produce an orthonormal basis from the given basis. a) { (1 , 1 , , 1) , (0 ,- 1 , 1 , 1) , (3 , , 1 , 1) } Solution: Denote the given basis by ~ z 1 = (1 , 1 , , 1), ~ z 2 = (0 ,- 1 , 1 , 1), ~ z 3 = (3 , , 1 , 1). Let ~w 1 = ~ z 1 . Then, we get ~w 2 = perp ~w 1 ( ~ z 2 ) = ~ z 2- proj ~w 1 ( ~ z 2 ) = ~ z 2- ~ z 2 ~w 1 k ~w 1 k 2 ~w 1 = (0 ,- 1 , 1 , 1)- 1 3 ( (0 ,- 1 , 1 , 1) (1 , 1 , , 1) ) (1 , 1 , , 1) = (0 ,- 1 , 1 , 1)- 3 (1 , 1 , , 1) = (0 ,- 1 , 1 , 1) Thus ~ z 2 was already orthogonal to ~ z 1 . Next we have ~w 3 = z 3- ~ z 3 ~w 1 k ~w 1 k 2 ~w 1- ( ~ z 3 ~w 2 ) k ~w 2 k 2 ~w 2 = (3 , , 1 , 1)- (3 , , 1 , 1) (1 , 1 , , 1) 3 (1 , 1 , , 1)- (3 , , 1 , 1) (0 ,- 1 , 1 , 1) 3 (0 ,- 1 , 1 , 1) = (3 , , 1 , 1)- 4 3 (1 , 1 , , 1)- 2 3 (0 ,- 1 , 1 , 1) = 1 3 (5 ,- 2 , 1 ,- 3) Thus, the set { ~w 1 , ~w 2 , ~w 3 } is an orthogonal basis for span { ~ z 1 ,~ z 2 ,~ z 3 } . To obtain an orthonor- mal basis, we normalize each vector to get 1 3 (1 , 1 , , 1) , 1 3 (0 ,- 1 , 1 , 1) , 1 39 (5 ,- 2 , 1 ,- 3) . b) { (1 , 1 , , 1 , 0) , (- 1 , , 1 , 1 , 1) , (1 , 1 , , 2 , 1) } Solution: Denote the given basis by ~ z 1 = (1 , 1 , , 1 , 0), ~ z 2 = (- 1 , , 1 , 1 , 1), ~ z 3 = (1 , 1 , , 2 , 1). Let ~w 1 = ~ z 1 . Then, we get ~w 2 = perp ~w 1 ( ~ z 2 ) = ~ z 2- proj ~w 1 ( ~ z 2 ) = ~ z 2- ~ z 2 ~w 1 k ~w 1 k 2 ~w 1 = (- 1 , , 1 , 1 , 1)- 1 3 ( (- 1 , , 1 , 1 , 1) (1 , 1 , , 1 , 0) ) (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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A5_soln - Math 235 Assignment 5 Solutions 1. Use the...

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