Let $ R $ be an integral domain.
Suppose you have $n$ vectors $ v_{1} , v_{2} , \ldots, v_{n} $ (not necessarily $ R$-linearly independent) in a free $R$-module $M$ of finite rank $m$. Let $N$ be the submodule of $M$ generated by $ v_{1} , v_{2} , ..., v_{n } $. Let $ \eta = \left \{ e_{1}, e_{2}, ..., e_{m} \right \} $ be a $ R $-basis of $ M$. Write the coordinate vectors of $ v_{i} $ with respect to $ \eta $ and concatenate these column vectors for each $ v_{i} $ to get a $ m \times n $ matrix $ A $. Let's call this matrix the "relations matrix of $ \left \{ v_{i} \right \}_{i=1}^{m} w.r.t. \eta $.
Let $P \in GL(m,R)$, $Q \in GL(n,R)$. Let $B = PAQ$. Is it true that there are generators $ w_{1} , w_{2} , ..., w_{n} $ of $N$, and a $ R $-basis $ \gamma = \left \{ f_{1} , f_{2} , ..., f_{m} \right \} $ of $ M $ such that $ B $ is the relations matrix of $ w_{i} $ w.r.t. $ \gamma $?