Dual of a linear representation

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

Suppose $G$ is a group and $\rho :G\to GL(V)$ is a linear representation of $G$.

Then the dual representation of $\rho$ of $G$ is given by $(\rho ^{*}(g)\theta )(v)=\theta (\rho (g^{-1})v)$ for $\theta \in V^{*}$, $g\in G$ and $v\in V$, where $V^{*}$ denotes the dual vector space of $V$.