Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v\_1,v\_2,v\_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a following quantitative statement:


Is there a finite number of triples in $\mathbb{Z}^{3}$ such that if the vectors $v\_1,v\_2,v\_3$ are linearly dependent, then for some triple $(k\_1,k\_2,k\_3)\neq 0$ we have
$$k\_{1}v\_{1}+k\_{2}v\_{2}+k\_{3}v\_{3}=0?$$ 


The important thing is that the collection should work for all triples of linearly dependent vectors under consideration. If the answer to this question is NO, can one then give a infinite collection of such "test" integer triples that would have a very small density in $\mathbb{Z}^{3}$? That is, we clearly do not need to consider all triples of integers - having considered (1,2,3), we do not need (2,4,6) in the collection etc. 



Linear independence of +/- 1 strings/vectors
There are only $2^3=8$ possibilities for the rows of $(v\_1 v\_2 v\_3)$. Duplicated rows don't matter. Hence there are only $2^8$ essentially different triples of vectors, so a collection of size at most $2^8$ suffices.


(I realise this is a bit trivial for an answer, but I don't have enough rep to comment)


