Given an integral symplectic matrix and a primitive vector, is their product also primitive?

Question

Given a matrix $A \in Sp(k,\mathbb{Z})$, and a column k-vector $g$ that is primitive ( $g \neq kr$ for any integer k and any column k-vector $r$), why does it follow that $Ag$ is also primitive? Can we take A from a larger space than the space of integral symplectic matrices?

Answer 1

Suppose $\,Ag\,$ is non-primitive, then $\,Ag=mr\,\,,\,\,m\in\mathbb{Z}\,\Longrightarrow g=mA^{-1}r\, $ , which means $\,g\,$ is not primitive