A5_soln - Math 235 Assignment 5 Solutions 1 Let B = 1 1 1 1...

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Unformatted text preview: Math 235 Assignment 5 Solutions 1. Let B = 1- 1 1 , , 1 1 1 1 , 2 1 1 and let S = Span( B ). a) Use the Gram-Schmidt process to produce an orthonormal basis for S . Solution: Denote the given basis by vector z 1 = 1- 1 1 , , vector z 2 = 1 1 1 1 , vector z 3 = 2 1 1 . Let vectorw 1 = vector z 1 . Then, we get vectorw 2 = vector z 2- proj vectorw 1 ( vector z 2 ) = vector z 2- vector z 2 · vectorw 1 bardbl vectorw 1 bardbl 2 vectorw 1 = 1 1 1 1 - 1 3 1- 1 1 , = 1 3 2 3 4 2 To simplify calculations we use vectorw 2 = 2 3 4 2 instead. Then, we get vectorw 3 = z 3- vector z 3 · vectorw 1 bardbl vectorw 1 bardbl 2 vectorw 1- ( vector z 3 · vectorw 2 ) bardbl vectorw 2 bardbl 2 vectorw 2 = 2 1 1 - 2 3 1- 1 1 , - 10 33 2 3 4 2 = 1 11 8- 10 5- 3 Thus, the set { vectorw...
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A5_soln - Math 235 Assignment 5 Solutions 1 Let B = 1 1 1 1...

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