General linear group of a vector space

Question

I am reading a text on Lie groups.

There is a whole chapter devoted to the group of invertible real or complex matrices of degree $n$, which are called the general linear groups (complex and real).

Further in the text, there is a reference to a general linear group of a vector space, in general. What is meant exactly by that? And why do we use the same terminology?

Answer 1

The set of bijective linear maps from a finite-dimensional vectorspace $V$ to itself is isomorphic (as a group) to the group of invertible $n\times n$ matrices, where $n$ is the dimension of the vectorspace (to get an isomorphism, just pick a basis).

The first one is what is meant by the general linear group of a vectorspace.