# hw4 - (iii Find min s ∈ S || v-s || i.e ﬁnd the minimum...

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Homework Assignment 4 - MAS 4105 - Fall 2002 NAME: CODEWORD: DUE : Tuesday, November 26 at the beginning of class. An extra point is awarded for submitting homework on time. This assignment will not be accepted after 5PM Wednesday, November 27. Instructions : Write in complete sentences using mathematical symbols where appropriate. Your reasoning should be clear. Write in such a way that the average student in the class can follow your work. It is OK for you to discuss this homework assignment with anyone, but you must acknowledge any assistance. It is NOT OK to copy anyone else’s work and it is NOT OK to copy from a book. Complete : On this assignment I obtained assistance from Date: Signed: 1. #1, p.247 2. #12, p.248 3. #13, p.248 4. Let S = Span((1 , 1 , 1 , 1) T , (1 , 0 , 1 , 0) T ) and let v = (1 , 2 , 3 , 4) T . (i) Find the orthogonal projection of v onto S by ﬁnding bases for S and S . (ii) Find the orthogonal projection of v onto S by solving the normal equations.

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Unformatted text preview: (iii) Find min s ∈ S || v-s || ; i.e. ﬁnd the minimum of || v-s || 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 i-2 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 1-3-2 3 , b = 1 1-5 , and S = Span((-2 , 1 ,-2) T , (0 ,-3 , 3) T ). (i) Compute the Gram-Schmidt QR-factorization 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 || b-s || ; ie. ﬁnd the minimum of || b-s || for s ∈ S . 2...
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hw4 - (iii Find min s ∈ S || v-s || i.e ﬁnd the minimum...

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