General linear group over a field as an algebraic group

Definition
Suppose $$k$$ is a field and $$n$$ is a natural number. The general linear group $$GL(n,k)$$ is defined as the group of all invertible $$n \times n$$ matrices over $$k$$. This has the natural structure of an algebraic group as described below:


 * The algebraic variety structures arises as follows: it is a Zariski-open subset of $$k^{n^2}$$, defined by the non-vanishing of the determinant polynomial.
 * The group structure is by matrix multiplication. Both the matrix multiplication and the inverse map are given globally by rational functions in the coordinates, and hence are morphisms of algebraic varieties.

General linear group of degree one as an algebraic group
For $$n = 1$$, the general linear group $$GL(1,k)$$ is the subset $$k^\ast$$ of the field $$k$$. Explicitly:

General linear group of degree two as an algebraic group
Here, we write a matrix $$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\\end{pmatrix}$$ as a bunch of four coordinates $$(x_{11}, x_{12}, x_{21}, x_{22})$$. Then: