Isomorphic linear representations

This article gives a basic definition in the following area: linear representation theory
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Definition

Suppose $V$ and $W$ are vector spaces over a field $K$.

Suppose $G$ is a group and $\rho :G\to GL(V)$, $\sigma :G\to GL(W)$ are linear representations of $G$.

$\rho$ and $\sigma$ are called isomorphic if there exists a map $\psi :V\to W$ a $K$-linear isomorphism of vector spaces such that $\sigma (g)=\psi \circ \rho (g)\circ \psi ^{-1}$.