Linear algebraic group

Remarks

Every linear algebraic group over $k$ is isomorphic to a closed subgroup scheme of $\mathrm{GL}_n$ for some $n$. So an equivalent definition is: a closed subgroup scheme of some general linear group $\mathrm{GL}_n$. In the classical language, with $k$ algebraically closed, it is a subgroup of $\mathrm{GL}_n(k)$ that is the zero set of a set of polynomials in the matrix entries. Many authors also require $G$ to be smooth.

Example: $\mathrm{SL}_n = \operatorname{Spec} k[x_{ij}] / (\det - 1)$. It is a closed subgroup of $\mathrm{GL}_n$, cut out by $\det = 1$, so it is affine.

Non-example: an elliptic curve $E$ over $k$ is an algebraic group, but it is projective of positive dimension and so not affine.