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 $\operatorname{GL}_k(V)$. 

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$.
The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties).