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problemset1 - The University of Texas at Austin Department...

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The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Problem Set One C. Caramanis Due: Wednesday, September 9, 2009. This problem set is intended to get us started thinking about differential equations and their solution, as well as properties of the solution. In addition, it will give a little practice for some basic linear algebra. 1. Exercise 1.2 from the book. 2. Using the Taylor expansion for sin, cos, and for the exponential, show (i.e., derive the rela- tionship, do not just quote the result) that e iαt = cos( αt ) + i sin( αt ) . Then based on this, conclude that for a C , the magnitude of x ( t ) = e at , depends on the real part of a , and not on the imaginary part of a . 3. Consider the three-by-three matrix 1 3 4 2 0 1 1 1 2 (a) Show that if vectors v 1 and v 2 both satisfy Av 1 = 0 and Av 2 = 0, then for any real numbers α and β , the vector v = αv 1 + βv 2 also satisfies: Av = 0. (b) Compute the set of vectors v such that Av = 0. (c) Computer the determinant of the matrix
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