Final-167D-F08

# Final-167D-F08 - Name Student ID Section 1 2 3 4 5 6 7 8...

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Name: Student ID: Section: 1 2 3 4 5 6 7 8 Total FINAL EXAM Math 167 Temple-F08

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Problem 1. (30pts) Let A be a real n × n symmetric matrix. (a) Prove that A has only real eigenvalues. (b) Assuming that A has n distinct eigenvalues, prove that A has an orthonormal basis of eigenvectors.
Problem 1. (Continued)

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Problem 2. (25pts) Let A = 1 2 1 2 1 2 0 . (a) Find the eigenvalues of A . (b) Find an orthonormal basis of eigenvectors of A. (c) Find an 2 × 2 orthogonal matrix S such that A = SDS T , where D is a diagonal matrix. (d) Find lim n →∞ A n .
Problem 2. (Continued)

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Problem 3. (20pts) Let A and B be m × n matrices with entries a ij and b ij , respectively. Prove that if Av i = Bv i for some basis { v 1 , ··· ,v n } of R n , then a ij = b ij for every i,j = 1 ··· n.
Problem 3. (Continued)

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Problem 4. (25pts) Assume that the populations y 1 and y 2 of two interacting species of animals evolves according to the equation y 0 ( t ) y 0 1 ( t
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Final-167D-F08 - Name Student ID Section 1 2 3 4 5 6 7 8...

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