We prove that whenever u,v∈N and A is a u× v matrix with integer entries and rank n, there is a u× n matrix B such that {A\vec k:\vec k∈Z^v}∩ N^u={B\vec x:\vec x∈Nⁿ}∩N^u. As a consequence we obtain the following result which answers a question of Hindman, Leader, and Strauss: Let R be a subring of the rationals with 1∈ R and let S={x∈ R:x>0}. If A is a finite matrix with rational entries, then there is a matrix B with no more columns than A such that the set of images of B in S via vectors with entries from S is exactly the same as as the set of images of A in S via vectors with entries from R.

We also show that the notion of image partition regularity is strictly stronger than that of weak image partition regularity in terms of Ramsey Theoretic consequences. That is, we show that for each u≧ 3, there are no v and a u× v matrix A such that for any \vec y∈ {A\vec k:\vec k∈Z^v}∩N^u, the set of entries of \vec y form (in some order) a length u arithmetic progression.

The first author acknowledges support received from the National Science Foundation (USA) via Grant DMS-1460023.