# Linear algebraic group

## Definition

### In terms of underlying variety

A linear algebraic group or affine algebraic group is an algebraic group where the underlying algebraic variety is an affine variety.

By a basic theorem of algebraic geometry, any affine algebraic geometry has a faithful linear representation, and can hence be realized as a linear algebraic group. Thus, we often view affine algebraic group and linear algebraic group as synonyms.

### In terms of embedding into general linear group

A linear algebraic group or affine algebraic group over a field $k$ is defined as an algebraic group $G$ such that there exists an embedding of $G$ as a closed subgroup of the general linear group $GL(n,k)$ for some choice of $n$ (and this embedding is a morphism of algebraic varieties).

Note that such an embedding as a closed subgroup of $GL(n,k)$ automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is because $GL(n,k)$ comes naturally equipped with an algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.