General linear group

Description

Let $R$ be a commutative ring and let $n \ge 1$ be an integer. The general linear group $GL_n(R)$ is the set of invertible $n \times n$ matrices over $R$, with matrix multiplication as operation. It is a group: the product of invertible matrices is invertible, the identity element is $I_n$, and the inverse of $A$ is $A^{-1}$. Equivalently, $GL_n(R)$ is the group of units of the matrix ring $M_n(R)$.

Remarks

For a field $F$, $GL_n(F)$ is isomorphic to the group of linear isomorphisms of $F^n$ to itself. For $n \ge 2$, the group $GL_n(F)$ is not abelian. For a finite field with $q$ elements, $|GL_n(\F_q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$. The special linear group is the subgroup of matrices with determinant $1$.

Explanations

L1 human

The general linear group collects all the invertible matrices of one size. We can multiply two of them, and we can undo each of them, so they form a group. Geometrically, its elements are all the ways to transform space linearly without flattening it.