📂Abstract Algebra

General Linear Group

Definition

The set of real invertible $n \times n$ matrices is denoted by $\mathrm{GL}(n, \mathbb{R})$ or $\mathrm{GL}_{n}(\mathbb{R})$ and is called the general linear group of degree $n$.

$$ \mathrm{GL}(n, \mathbb{R}) := \left\{ n \times n \text{ invertible matrix} \right\} = M_{n \times n}(\mathbb{R}) \setminus {\left\{ A \in M_{n \times n}(\mathbb{R}) : \det{A} = 0 \right\}} $$

Explanation

Since it consists only of invertible matrices, it forms a group with respect to matrix multiplication. Moreover, it has a differentiable structure, making it a Lie group.