Let $x$ be a $\mathbb{R}^n$ valued path with finite variation on $[0,T]$ and $A$ be a matrix of $d\times n$. It is easy to see that $Ax$ is a path with finite variation.
Let $S^N(x)_{0,1}$ be the step-N signature of $x$ on $[0,1]$.
What is the relationship between $S^N(x)_{0,1}$ and $S^N(Ax)_{0,1}$?