Dimension of an algebraic group

Definition
The dimension of an algebraic group over a field is defined in the following equivalent ways:

Facts

 * The dimension is an invariant under any isomorphism of algebraic groups. It is possible, however, to have the same abstract group arising as algebraic groups of different dimensions over a given field.
 * Given an algebraic group $$G$$ over a field extension $$L$$ of $$K$$, $$G$$ naturally acquires the structure of an algebraic group over $$K$$. The dimension of $$G$$ over $$K$$ is the product of the dimension of $$G$$ over $$L$$ and the degree of the extension $$L/K$$.
 * The dimension of an algebraic group equals the dimension of its connected component of identity.

Particular cases

 * The trivial group, viewed as an algebraic group over any field, is zero-dimensional.
 * The additive group of a field, as well as the multiplicative group of a field, are both one-dimensional as algebraic groups under the usual structure.
 * The additive group of a $$n$$-dimensional vector space has dimension $$n$$ as an algebraic group.
 * The general linear group over a field of degree $$n$$ has dimension $$n^2$$ as an algebraic group.