Let $Ax=0$ a linear system of equations over the integers (i.e. all the entries of $A$ and $x$ are integers), where $A$ is an $m \times n $ matrix.
Do we know that if $n>m$ then there is a nonzero solution over the integers?
In linear algebra, for a homogeneous system this follows by the rank-nullity theorem. There has to be at least one linearly independent column, making the dimension of the kernel nontrivial. Is there something equivalent to this when we are no longer over a field, but instead over $Z$? Or is there perhaps some other useful fact about systems of linear equations over the integers that helps guarantee a solution to such a homogeneous system?