601-hw-v - MATH 601, AUTUMN 2008 Additional homework V,...

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Additional homework V, October 30 PROBLEMS 1. Find the solution v ( t ) = ( x ( t ) , y ( t ) , z ( t )) to the system of ordinary differential equations ˙ v = Tv with the initial condition v (0) = (1 , 1 , 1), where T : R 3 R 3 is the operator T x y z = 1 - 2 2 1 - 1 3 0 0 2 · x y z , with vectors of R 3 written in the column form. 2. Find an explicit formula for the solutions x ( t ) , y ( t )) to the system of ordinary differential equations ˙ x = 3 x - 9 y , ˙ y = x - 3 y with any given initial conditions x (0) = a , y (0) = b , a, b C . 3. Let T End R 4 be the linear operator given by T x y z u = - 1 1 1 - 2 0 0 - 1 0 0 0 1 0 0 0 0 1 · x y z u , with vectors of R 4 written in the column form. Find all eigenvalues of T and construct a basis for each eigenspace of T . 4. Evaluate T 38 w for w = (1 , 1 , 1) R 3 and the operator T : R 3 R 3 defined by T x y z = 15 40 - 17 10 25 - 11 36 92 - 40 · x y z . (No calculators, please!)
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This note was uploaded on 04/18/2009 for the course MATH 601 taught by Professor Un during the Fall '08 term at Ohio State.

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601-hw-v - MATH 601, AUTUMN 2008 Additional homework V,...

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