In J. Meier's book "groups, graphs and trees" after remark 3.8 it is stated that
A group that can be faithfully represented as a matrix group is called a linear group.
Other sources (most of the time) stated
A group is linear if it admits a faithful representation of $\operatorname{GL}(n,\mathbb C)$.
In other cases I've seen them stating the second definition over any field $\mathbb K$. But even in cases where the authors speficially mentioned $\operatorname{GL}(n,\mathbb C)$, they still considered groups to be linear if they admitted a faithful representation e.g. of $\operatorname{GL}(n,\mathbb Z)$. So I'm a bit confused.
Question 1: Suppose we found a faithful representation like $G\to \operatorname{Aut}(\mathbb Z^n)$, then $\operatorname{Aut}(\mathbb Z^n) \cong \operatorname{GL}(n,\mathbb Z)$. Does this suffice to conclude that $G$ is linear? I suppose that this is not sufficient, but I can't find any reference in any of my books.
Thank you very much for any help!
It is a famous result of L. Auslander that any polycyclic group admits a faithful representation to $GL_n(\Bbb Z)$ for some $n\ge 1$. This is much more than being linear. Not every linear group has such a faithful representation.
I suspect that $GL_n(\Bbb Z)$ arises in a certain context in your text, which is different from just linear groups.