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.