Let $k$ be a field, let $n\in\mathbb{N}$, let $V$ be an $n$-dimensional [vector space](/page/Vector%20Space) over $k$, and let $\mathcal{B}=(v_1,\dots,v_n)$ be an ordered basis of $V$. For each $T\in GL(V)$, let $[T]_{\mathcal{B}}\in M_n(k)$ denote the matrix of the $k$-[linear map](/page/Linear%20Map) $T:V\to V$ with respect to the ordered basis $\mathcal{B}$. Then the map $\Phi_{\mathcal{B}}:GL(V)\to GL_n(k)$ defined by

\begin{align*} \Phi_{\mathcal{B}}(T)=[T]_{\mathcal{B}} \end{align*}

is an isomorphism of groups.