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General Linear Group 📂Abstract Algebra

General Linear Group

Definition

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

GL(n,R):={n×n invertible matrix}=Mn×n(R)∖{A∈Mn×n(R):det⁡A=0} \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.