First let me give you some definitions for the clarification of the problem:(adsbygoogle = window.adsbygoogle || []).push({});

Definition (Roman, Linear Algebra)

A nonempty set S of a vector space V with a field F is linearly independent if for any distinct vectors [itex]s_{1} , \dots , s_{n}[/itex] for all [itex]a_{1} , \dots , a_{n}[/itex] if [itex]a_{1}u_{1} + \cdots + a_{n}u{n} = 0 [/itex] then [itex]a_{1} = \cdots = a_{n} = 0[/itex].

But by this definition, I don't think I can handle an infinite set S (but you may!!). When S is infinite, the n up there should be infinite too. Then everything just becomes confusing to me. Can n actually be inifinite? Isn't it an error in the sense of the first order theory? Shouldn't the definition be modifed somewhat to be more rigorous, and to be fitter to deal with the set S in the case where it is infinite? Shouldn't there be any problem of countability of n when S is infinite?

I can't even properly state what actually I don't know. Please give me your helpful comment. Thanks.

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# How to prove the linear indenpendency of an infinite set

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