Definition:Equivalent Linear Representations

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

Consider two representations $\rho: G \to \operatorname{GL} \left({V}\right)$ and $\rho \, ': G \to \operatorname{GL} \left({W}\right)$ of $G$.

Then $\rho$ and $\rho \, '$ are called equivalent (representations) iff their correspondent $G$-modules using Equivalence of Representation Definitions are isomorphic.