Matrices $A_1, A_2 \in \operatorname{GL}_n(\mathbb{F})$ represent the same element of $\operatorname{GL}(V)$ with respect to two different bases if and only if they are conjugate, i.e.\ there exists $X \in \operatorname{GL}_n(\mathbb{F})$ such that
\begin{align*}
A_2 = X A_1 X^{-1}.
\end{align*}