Solving Rational Dependence in Vector Spaces

[bndnchrs]

Hi guys:

I've got a problem I've been working on for some weeks and this might be the key to unlocking it.

The question is:

Given a vector in R^k, what is the measure of the set of vectors whose components are rationally dependent?

Rationally dependent means for a given vector, you may find a vector with rational coefficients such that their inner product is 0.

(1/2,1/3,1/6) is RD because of (2,3,-12), for example.

 

[Landau]

(1/2,1/3,1/6) is RD because of (2,3,-12), for example.

But this is easily generalized: every 3-tuple (p,q,r) with rational entries p,q,r is RD because
$$(p,q,r)\cdot (1/p,1/q,-2/r)=0$$
and 1/p,1/q,-2/r are rational. This is then easily generalized to R^n instead of R^3. Hence the set of RD vectors contains then ones with rational entries. But the latter one is already dense in R^n.

 

[bndnchrs]

I suppose it was late, and this meant I had to improperly state the question!

Really, the question is does the set of RD vectors have nonzero measure over R^k, not whether they are dense or not. Of course the rationals are rationally dependent and dense, but they are a set of measure zero in R. So then the issue is whether a.a. collections of irrationals are RD, and my intuition says no.

So RID vectors requires k-1 components to be irrational. I don't have a hold on determining exactly "how much" of these sets are RD.

 

[tmccullough]

I think I understand the statement, but I'm not totally sure. I suppose that you mean what is the measure of the set of all vectors which are rationally dependent (if this is even measurable)? I've never heard of rationally dependent before...

I think this is the answer:
Suppose $r\in\mathbb{R}^n$. Note that the measure of the set of all vectors perpendicular to $r$ is zero. Now, what is the measure of a countable union of sets of measure zero?

Does that help?

 

[Office_Shredder]

I think I understand the statement, but I'm not totally sure. I suppose that you mean what is the measure of the set of all vectors which are rationally dependent (if this is even measurable)? I've never heard of rationally dependent before...

I think this is the answer:
Suppose $r\in\mathbb{R}^n$. Note that the measure of the set of all vectors perpendicular to $r$ is zero. Now, what is the measure of a countable union of sets of measure zero?

Does that help?

I don't see why you're only taking a countable union. There are certainly uncountably many vectors in Rⁿ.A modified argument seems to work though. Pick a vector with rational components q in Qⁿ. Any rationally dependent vector is perpendicular to some such q, so lies in one of countably many measure zero sets

 

[tmccullough]

That's not modified at all, that is the argument. I didn't want to give everything away.

 

[bndnchrs]

Much appreciated. I believe this solves my question.

Don't worry about revealing the "proof", I would say that this problem is just a redefinition of a small mechanism in a larger problem, which has nothing to do with linear algebra, actually, so letting me in on the mechanism is of no great detriment to my progress.

Thank you both!