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MAD 4401 Review for Test 3 James Keesling April 10, 2009 1 Linear Ordinary Diﬀerential Equations Problem 1.1. Let A be an n × n matrix. Deﬁne exp( tA ) . Problem 1.2. Let A = ± 1 2 1 0 ² and compute exp( tA ) . Problem 1.3. Let x ( t ) = x 1 ( t ) x 2 ( t ) . . . x n ( t ) and suppose that A is an n × n matrix with constant entries. Consider the following diﬀerential equation. dx dt = Ax Show that the general solution of this equation is given by the formula x ( t ) = exp( tA ) · C 1 . . . C n . Problem 1.4. Solve the following diﬀerential equation using matrix methods. d 2 x dt 2 + 2 dx dt - 3 x = 0 Problem 1.5. Solve the following diﬀerential equation dx dt = Ax for the following A . A = ± 1 1 0 2 ² 1

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2 Vector and Matrix Norms Problem 2.1. Give the deﬁnition of a norm on an n - dimensional vector space. Problem 2.2. For any two norms || x || 1 and || x || 2 on the n - dimensional vector space V there are positive numbers r
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