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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.

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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)

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  • $\begingroup$ Indeed. I will post a refined version of the question as well as $2^n$ is too much for me. $\endgroup$ – TOM May 7 '15 at 3:25
  • $\begingroup$ Also, did you not mean that $\binom{2^n}{3}$ is an upper bounds as there are these many triples of vectors whose linear independence we have to check? $\endgroup$ – TOM May 7 '15 at 4:16
  • $\begingroup$ No. The point is that however many rows there are (i.e. however large $n$ is), we only only care about which of the $8$ possible rows are present for the triple $(v_1 v_2 v_3)$, since we don't care about multiplicity of rows. This gives a bound of $2^8$, not $2^n$. $\endgroup$ – David May 7 '15 at 12:19

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