204Final082106 - equation with the initial conditions y 1...

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Economics 204–Final Exam–August 21, 2006, 2-5pm Each question is worth 25% of the total Please use separate bluebooks for each of the two Parts Part I 1. Prove that if two vector spaces X and Y over the same Feld F are isomorphic, then dim X =d im Y . 2. Consider the di±erential equation y 1 y 2 ! 0 = 14 4 1 ! y 1 y 2 ! (a) Compute the eigenvalues of the matrix. Briefly discuss what this tells you about the qualitative nature of the solutions of the di±erential equation. (b) Explain how we know there must be an orthonormal basis of R 2 composed of eigenvectors of the matrix. Compute this orthonormal basis. (c) Sketch the qualitative behavior of the solutions of the equation in a phase plane diagram. (d) ²ind a solution of the Initial Value Problem formed by combining the di±erential
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Unformatted text preview: equation with the initial conditions y 1 (0) = A,y 2 (0) = B . Is the solution of the Initial Value Problem unique? Part II 3. Let Ψ be a correspondence from X to Y which is compact-valued and upper hemi-continuous, C a compact subset of X . Let Ψ( C ) = ∪ x ∈ C Ψ( x ). Prove that Ψ( C ) is compact. ²or full credit, you must use the open set deFnitions of compactness and upper hemicontinuity; a correct proof using the sequential formulations of compactness and upper hemicontinuity will receive three-fourths credit. 4. Prove that for all n ∈ N , n X k =1 k = n ( n + 1) 2 1...
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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