Definition:Linear Representation

Definition
Let $\left({\mathbb k, +, \circ}\right)$ be a field.

Let $V$ be a vector space over $\mathbb k$ of finite dimension.

Let $\operatorname {GL} \left({V}\right)$ be the general linear group of $V$.

Let $\left({G, \cdot}\right)$ be a finite group.

A linear representation of $G$ on $V$ is a group homomorphism $\rho: G \to \operatorname {GL} \left({V}\right)$.

Also see
By Equivalence of Representation Definitions, a linear representation of $G$ on $V$ is completely specified by a linear action of $G$ on $V$.