Definition talk:Linearly Independent
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Equivalent Definition
Is this definition distinct enough to merit its own "alternative definition"? :
A set is linearly dependent if one vector in it can be written as a linear combination of the others
A set is linearly independent if no vector can be written as a linear combination of any others
I'm inclined to say yes, but I'm not sure how to present it, as I only know how this stuff works in $\R^n$
--GFauxPas 21:40, 25 March 2012 (EDT)
- If I'm not mistaken, what you just wrote is equivalent to the definition on the page if and only if $R$ is a division ring. Comments? –Abcxyz (talk | contribs) 23:16, 25 March 2012 (EDT)
- If $R$ is a division ring, then a unitary $R$-module is in fact a vector space. Would it be a good idea to have a page like this for vector spaces?
- As for your "alternative definition", I think that somebody else (other than me) should say whether it should go on to its own page or not. –Abcxyz (talk | contribs) 00:12, 26 March 2012 (EDT)\
- Turns out PW already has this as a theorem. You're right about the limitation, and as such I think it's best left as a theorem because the one on the page is more general. --GFauxPas 10:30, 26 March 2012 (EDT)