Let $k$ be a field, let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $k$, and let $n=\dim_k V$. Let $\operatorname{Fr}(V)$ denote the set of ordered basis frames of $V$, meaning ordered $n$-tuples $\mathcal{B}=(v_1,\dots,v_n)$ that form a basis of $V$; when $n=0$, this is the unique empty ordered basis. The map

\begin{align*} GL(V)\times \operatorname{Fr}(V)\to \operatorname{Fr}(V), \qquad (T,(v_1,\dots,v_n))\mapsto (T(v_1),\dots,T(v_n)) \end{align*}

defines a simply transitive left [group action](/page/Group%20Action). Equivalently, for any two ordered bases $\mathcal{B},\mathcal{C}\in \operatorname{Fr}(V)$, there exists a unique $T\in GL(V)$ such that $T\mathcal{B}=\mathcal{C}$.