In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.