Let x(1), ... , x(m) be solutions of x = P(t)x on the interval < t < . Assume that P

Chapter 7, Problem 8

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Let x(1), ... , x(m) be solutions of x = P(t)x on the interval < t < . Assume that P iscontinuous, and let t0 be an arbitrary point in the given interval. Show that x(1), ... , x(m)are linearly dependent for < t < if (and only if) x(1)(t0), ... , x(m)(t0) are linearly dependent.In other words x(1), ... , x(m) are linearly dependent on the interval (, ) if they arelinearly dependent at any point in it.Hint: There are constants c1, ... , cm that satisfy c1x(1)(t0) ++ cmx(m)(t0) = 0. Letz(t) = c1x(1)(t) ++ cmx(m)(t), and use the uniqueness theorem to show that z(t) = 0for each t in < t < .