General linear group ↩

shorthands: {}
aliases: [GL, groupe linéiaire, general linear group, Groupe linéaire]
created: 2021-10-18 18:02:38
modified: 2022-01-10 04:13:04

Motivation and abstract definition

Let be an associative algebra with unit element . Then the set of all units in form a group under multiplication.

Now if we look at the algebra formed by the endomorphisms of a vector space, , we see that it is associative, since function composition is associative by definition. We also have a group of units in . We denote it by and call it the general linear group over .

Less formal but more useful definition

field and integer, then is the group of invertible matrices over the field , with the product operation being the usual matrix dot product. Usually .

For a vector space : is the group of invertible linear transformations with the product operation being the usual "do the right transformation then the left one". Usually is a linear space over the complex field .

If is finite dimensional, let's say , then .