Definition:Faithful Linear Representation of Group

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

Let $V$ be a vector space over $k$ 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.

Let $\rho : G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$.

Then $\rho$ is faithful if the kernel of $\rho$ is trivial.