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SampleExam1 - Problem 1 a For a system of linear equations...

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Unformatted text preview: Problem 1: a) For a system of linear equations AX :: c, What must be true for the solution X to be unique? b) Complete the statement: The eigenvectors of a symmetric matrix With repeated eigenvalues are (always, sometimes, or never) mutually linearly inde- pendent. c) Under What conditions do you expect to obtain the Jordan canonical form when seeking to diagonalize a matrix A? d) Obtain the self—adjoint form of the differential operator Problem 2: Obtain the solution to the following system of differential equations using matrix operations 61 % = 2561 + $3: diL'g _ 1: dt ” 2’ day *8; 1 £62 + 3333, Problem 3: Consider the one—dimensional, unsteady diffusion equation 8U (92” —— : —— < < (9t @8902) 0 " $ ‘“ g with the boundary conditions u(0,t) = 0, u(€,t) = 0, and the initial condition M220) = fix)- Using the method of separation of variables, obtain the solution for Mac, t) in terms of an eigenfunction expansion. ...
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