Definition:Linear Representation

Linear Representation of a Finite Group
Let $(k,+,\circ)$ be a field.

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

Let $GL(V)$ be the general linear group of $V$.

Let $(G, \cdot)$ be a finite group.

A linear representation of $G$ on $V$ is a homomorphism of groups $\rho : G \to GL(V)$.

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