Math 218- Final exam- Fall 08-09

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

This preview shows page 1. Sign up to view the full content.

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

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)...
View Full Document

## 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.

Ask a homework question - tutors are online