General linear group over reals

Definition
The general linear group over reals of degree $$n$$, denoted $$GL(n,\R)$$ or $$GL_n(\R)$$, is defined as the general linear group of degree $$n$$ over the field of real numbers $$\R$$.

Some of the properties of these general linear groups generalize to general linear groups over fields that resemble the reals in one or more important respect: for instance, formally real fields, totally real fields, ordered fields, Pythagorean fields, and quadratically closed fields.

Structures
Each group $$GL(n,\R)$$ can be thought of in any of the following ways:


 * It is a real Lie group.
 * It is a linear algebraic group over the field of real numbers (note that this is not an algebraically closed field).
 * It is a topological group.

Topological/Lie group properties
The topology here is the subspace topology from the Euclidean topology on the set of all matrices, which is identified with the Euclidean space $$\R^4$$.