practiceprob5_soln

practiceprob5_soln - Math 235 Assignment 5 Practice...

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Math 235 Assignment 5 Practice Problems Solutions 1. Consider the subspace S = Span 1 0 - 1 1 , , 1 1 1 1 , 2 0 1 1 of M 2 × 2 ( R ). (a) Use the Gram-Schmidt process to produce an orthogonal basis for S . Solution: Denote the given basis by ~ z 1 = 1 0 - 1 1 , , ~ z 2 = 1 1 1 1 , ~ z 3 = 2 0 1 1 . Let ~w 1 = ~ z 1 . Then, we get ~w 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 0 - 1 1 , = 1 3 2 3 4 2 To simplify calculations we use ~w 2 = 2 3 4 2 instead. Then, we get ~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 = 2 0 1 1 - 2 3 1 0 - 1 1 , - 10 33 2 3 4 2 = 1 11 8 - 10 5 - 3 We pick ~w 3 = 8 - 10 5 - 3 . Then the set { ~w 1 , ~w 2 , ~w 3 } is an orthogonal basis for Span { ~ z 1 , ~ z 2 , ~ z 3 } .
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