Definition talk:Linearly Independent

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)