I encounter the following problem.
Let A be an m*n matrix with all integer entries (may be positive or nonpositive).
Suppose that Ax=0 admits a nonnegative solution x (i.e., xi>=0 for all i=1,...n).
Is it true that A also admits a nonnegative solution with all entries being rational numbers?
I know the answer is yes when we take away "nonnegative" from the above: putting A 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.
but how about with "nonnegative" added in?