exam1f11

exam1f11 - Foundations of Computational Math I Exam 1...

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Foundations of Computational Math I Exam 1 Take-home Exam Open Notes, Textbook, Homework Solutions Only Calculators Allowed No collaborations with anyone Due beginning of Class Wednesday, October 26, 2011 Question Points Points Possible Awarded 1. Basics 25 2. Linear operators 25 3. Floating point 25 4. Factorization 25 5. Orthogonal 25 Factorization Total 125 Points Name: Alias: to be used when posting anonymous grade list. 1
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Problem 1 (25 points) 1.a (10 points) Suppose A R m × n and consider the matrix 2-norm k A k 2 = max k x k 2 =1 k Ax k 2 Show that k A k 2 ≥ k A 1 k 2 where A = ± A 1 A 2 ² , m = m 1 + m 2 , A 1 R m 1 × n , and A 2 R m 2 × n . 2
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1.b (15 points) Let S 1 R n and S 2 R n be two subspaces of R n . (i) (5 points) – Suppose x 1 ∈ S 1 , x 1 / ∈ S 1 ∩ S 2 . x 2 ∈ S 2 , and x 2 / ∈ S 1 ∩ S 2 . Show that x 1 and x 2 are linearly independent. 3
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(ii) (10 points) – Suppose x 1 ∈ S 1 , x 1 / ∈ S 1 ∩S 2 . x 2 ∈ S 2 , and x 2 / ∈ S 1 ∩S 2 . Also, suppose that x 3 ∈ S 1 ∩ S 2 and x 3 6 = 0, i.e., the intersection is not empty. Show that x 1 , x 2 and x 3 are linearly independent. (Note the result of the previous part of the problem may be useful.) 4
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Problem 2 (25 points) 2.a (15 points) Recall that P n , the set of polynomials of degree less than or equal to n , and the operation
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exam1f11 - Foundations of Computational Math I Exam 1...

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