Rational solutions of homogeneous equations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:05:39Z http://mathoverflow.net/feeds/question/38155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38155/rational-solutions-of-homogeneous-equations Rational solutions of homogeneous equations thku 2010-09-09T08:16:59Z 2010-09-09T08:38:26Z <p>Can every solution of a homogeneous linear system be approximated by a solution in rational numbers? </p> <p>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 $&lt; \epsilon$ from $x$?</p> <p>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.</p> http://mathoverflow.net/questions/38155/rational-solutions-of-homogeneous-equations/38156#38156 Answer by Ricky Demer for Rational solutions of homogeneous equations Ricky Demer 2010-09-09T08:30:02Z 2010-09-09T08:30:02Z <p>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</p>