Definition:Linear Representation

Definition (Groups)
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 $\rho: G \to \operatorname {GL} \left({V}\right)$.

Definition (Algebras)
Let $K$ be a field.

Let $A$ be an associative, unitary algebra over $K$.

Then a representation of $A$ is a vector space $V$ over $K$ equipped with a homomorphism of algebras


 * $\rho:A\rightarrow \operatorname{End}(V)$

Note that a group representation is the same as a $K[G]$-representation.

Also see
By Existence of Bijection between Linear Group Action and Linear Representation, a linear representation of $G$ on $V$ is completely specified by a linear action of $G$ on $V$.