In this problem we indicate how to show that u(t) and v(t), as given by Eqs. (17), are

Chapter 7, Problem 27

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In this problem we indicate how to show that u(t) and v(t), as given by Eqs. (17), are linearly independent. Let r1 = + i and r1 = i be a pair of conjugate eigenvalues of the coefficient matrix A of Eq. (1); let (1) = a + ib and (1) = a ib be the corresponding eigenvectors. Recall that it was stated in Section 7.3 that if r1 = r1, then (1) and (1) are linearly independent. (a) First we show that a and b are linearly independent. Consider the equation c1a + c2b = 0. Express a and b in terms of (1) and (1) , and then show that (c1 ic2) (1) + (c1 + ic2) (1) = 0. (b) Show that c1 ic2 = 0 and c1 + ic2 = 0 and then that c1 = 0 and c2 = 0. Consequently, a and b are linearly independent. (c) To show that u(t) and v(t) are linearly independent, consider the equation c1u(t0) + c2v(t0) = 0, where t0 is an arbitrary point. Rewrite this equation in terms of a and b, and then proceed as in part (b) to show that c1 = 0 and c2 = 0. Hence u(t) and v(t) are linearly independent at the arbitrary point t0. Therefore they are linearly independent at every point and on every interval.