Linear Representations of Groups

Definition

Suppose $G$ is a finite group with identity 1 and suppose $V$ is a vector space over the field $\C $ of complex numbers. A linear representation $\rho : G \to GL(V)$ of $G$ in $V$ is a group homomorphism from $G$ to the general linear group $GL(V)$. Given $\rho $, we call $V$ a representation space (or representation) of $G$

Suppose $V$ has finite dimension $n$. In this case, we call $n$ the degree of the representation $\rho $. Given a basis $e_1, \dots , e_n$ of $V$,