exam1f11

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

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 15

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online