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I was studying the book of ODE of Simmons, and he says that the Wronskian ($W$) is $0$ iff the solutions are LI. Even he present a proof, a counterexample that $W = 0$ does not implies LD is $ \left| x \right|, \ x $ and he used that to prove the theorem, but since it is not well tested I can not follow, in addition to not really understand what he did.

Anyway, I want to know some proof of this fact, the fact that the solutions form a vector space (easy) of dimension $n$ (difficult to me T_T).

Thanks.

If someone knows some book to study ODE (for a beginner), I'll appreciate it.

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    @anon, of cours$e$ "lin$e$arly d$e$pendent". It $w$as not as $e$asy to figure out because it is not clearly a predicate where it appears.2011-09-25

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You might take a look at Birkhoff and Rota, "Ordinary Differential Equations". It treats the Wronskian in section II.3.