Question

Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?

In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\times n$-matrix $A$ (possibly $m>n$) with integer entries (say all entries $1,0,-1$ for simplicity). Given a solution $x\in {\Bbb R}^n$ and $\epsilon>0$, do there exist solutions in ${\Bbb Q}^n$ within distance $< \epsilon$ from $x$?

I am sure this kind of question has been considered somewhere. However, as a topologist, I have no idea where to look this up. Apart from answers also hints to literature about this genre of questions would be appreciated.

Answer 1

Since A has integer entries, putting it in reduced row-echelon form shows that the solution-space is spanned by vectors with rational coordinates. Rational multiples of the spanning vectors are then dense in the solution-space, so vectors with rational coordinates are also dense in the solution-space. Therefore every solution of a homogeneous linear system can be approximated by a solution in the rational numbers. QED