Math 218- Final exam- Fall 08-09

01 10 exercise 6 16 points 3 points 13 points 0 1

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Unformatted text preview: se 6: 16 points (3 points + 13 points) ∞. 0, 1 and 1 with corresponding Find Consider the matrix 3 . 100 1 2 0. 123 a) Justify that the matrix A has a QR-decomposition. b) Find the matrices Q and R of this decomposition. Exercise 7: 24 points (4 points for each question) , 0 for every weighted inner product on a) Find two distinct non-zero vectors u and v in R2, such that 2 R. b) Find two distinct non-zero vectors u and v in R2, such that , 0 for any inner product on R2. ,…, is an orthonormal basis of a real inner product space V. Show that for every , c) Suppose that we have , , . d) Let u and v be vectors in an inner product space. Prove that if and only if and are orthogonal. e) Let A be a square matrix such that . What can you say about the eigenvalues of A? f) Show that the matrix √ √ √ is orthogonal. √ Definition: A proper orthogonal matrix is an orthogonal matrix of which the determinant is equal to 1. Check that P is not proper and deduce from P a proper orthogonal matrix P’. (Use a relevant small change only)...
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This note was uploaded on 02/07/2014 for the course MATH 218 taught by Professor Rananassif during the Summer '07 term at American University of Beirut.

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