Consider a 2 2 system x_ = Ax. If we assume that r1 _= r2, the general solution is x =

Chapter 7, Problem 20

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Consider a 2 2 system x_ = Ax. If we assume that r1 _= r2, the general solution is x = c1 (1) er1t +c2 (2) er2t , provided that (1) and (2) are linearly independent. In this problem we establish the linear independence of (1) and (2) by assuming that they are linearly dependent and then showing that this leads to a contradiction. a. Explain how we know that (1) satisfies the matrix equation (A r1I) (1) = 0; similarly, explain why (A r2I) (2) = 0. b. Show that (A r2I) (1) = (r1 r2) (1) . c. Suppose that (1) and (2) are linearly dependent. Then c1 (1) + c2 (2) = 0 and at least one of c1 and c2 (say, c1) is not zero. Show that (A r2I)(c1 (1) + c2 (2) ) = 0, and also show that (Ar2I)(c1 (1) +c2 (2) ) = c1(r1 r2) (1) . Hence c1 = 0, which is a contradiction. Therefore, (1) and (2) are linearly independent. d. Modify the argument of part c if we assume that c2 _= 0. e. Carry out a similar argument for the case A is 33; note that the procedure can be extended to an arbitrary value of n. 2