Why is the name general “linear” group?

Question

Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?

Answer 1

$GL(V)$ is the group of linear transformations over a vector space $V$. You can also, as you have, write it $GL_n(K)$ if $V$ is an $n$-dimensional vector space over a field $K$, and thus isomorphic to $K^n$.

So, the "linear" part refers to the linearity property of the transformations: given vectors $v,w\in V$, scalars $\alpha,\beta\in K$, and a transformation $T\in GL(V)$, $$T(\alpha v+\beta w)=\alpha T(v) + \beta T(w).$$

Answer 2

The term linear here refers to the fact that it is a group consisting of linear transformations of some vector space. In some sense, all groups are "linear" like this, but usually if one refers to something as a linear group, then a specific realization as a group of linear transformations is usually (at least implicitly) meant.