In this paper linear representations of finite groups are introduced, and the associated character theory with it. Some work of linear representations of the dihedral group $D_n$ and the symmetric group $S_n$ is presented. \\We also take a look at the finite matrix groups $\textbf{GL}(\mathbb{F}_q)$ and $\textbf{SL}(\mathbb{F}_q)$. The character table for $\textbf{SL}(\mathbb{F}_4)$ and its representation spaces in an implicit form are calculated. We define the standard representation $\varphi $ of $\textbf{SL}(\mathbb{F}_q)$ and prove that it is irreducible for an arbitrary finite field $\mathbb{F}_q$.