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Putnam 1995 A5

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July 1, 2014
Putnamfunctionlimitvectorlinear algebramatrixcollege contests

Problem Statement

Let x1,x2,⋯ ,xnx_1,x_2,\cdots, x_n be real valued differentiable functions of a variable tt which satisfy \begin{align*} & \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ & \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ & \;\qquad \vdots \\ & \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \end{align*} For some constants aij>0a_{ij}>0. Suppose that lim⁡t→∞xi(t)=0\lim_{t \to \infty}x_i(t)=0 for all 1≤i≤n1\le i \le n. Are the functions xix_i necessarily linearly dependent?
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