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Unformatted text preview: (iii) Find min s ∈ S  vs  ; i.e. ﬁnd the minimum of  vs  where s ∈ S . 5. #5, p.259 6. #12, p.260 7. #24, p.269 8. As usual let e 1 , e 2 , ... , e n denote the columns of the n × n identity matrix I . Let 1 = (1 , 1 ,..., 1) T . For 1 ≤ i ≤ n , deﬁne v i := e i2 n 1 . Let Q = [ v 1 v 2 ··· v n ] (i.e. the matrix whose columns are v 1 ,v 2 ,...,v n ). Prove that Q is an orthogonal matrix. 9. #25, p.289 1 10. Let A = 2 132 3 , b = 1 15 , and S = Span((2 , 1 ,2) T , (0 ,3 , 3) T ). (i) Compute the GramSchmidt QRfactorization of A . (ii) Solve the normal equations A T A x = A T b . (iii) Find the orthogonal projection p of b onto S . (iv) Find min s ∈ S  bs  ; ie. ﬁnd the minimum of  bs  for s ∈ S . 2...
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 Spring '09
 RUDYAK
 Linear Algebra, orthogonal projection

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