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Unformatted text preview: MATH 224 PROBLEM SET 5 DUE: 26 FEBRUARY 2008 Reading. § 4.10–4.11. Problems from the book: • 4.9.4, 4.9.6 • 4.10.2, 4.10.4, 4.10.5, 4.10.12, 4.10.13, 4.10.14 • 4.11.1, 4.11.2 Additional problems: 1. Compute integraltextintegraltext R ( x + y ) e x 2- y 2 dA , where R is the region bounded by the lines x- y = , x- y = 2 , x + y = and x + y = 3 . 2. Evaluate integraltextintegraltextintegraltext R xyz | dx dy dz | over the region R between the spheres centered at the origin of radius r = 2 and r = 4 , and above the cone φ = π 3 . 3. Recall that a matrix is orthogonal if and only if AA T = I . Let O ( n ) denote the set of n × n orthogonal matrices. a. Show that O ( n ) is a group. b. Let A ∈ O ( n ) and let χ A ( t ) denote its characteristic polynomial. Suppose that α ∈ R is a root of χ A ( t ) . Prove that α = ± 1 . c. We have seen in class that a permutation matrix M σ is an orthogonal matrix, for every σ ∈ S n . Can you describe all of the roots of the characteristic poly-....
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This homework help was uploaded on 02/24/2008 for the course MATH 2240 taught by Professor Taraholm during the Spring '08 term at Cornell.
- Spring '08