Given 2 vectors ($\hat u$, $\hat v$) and their representations in two right-handed $\mathbb R^3$ orthonormal bases ($\textbf M$, $\textbf N$), how do you construct the transformation matrix, $\textbf A_{MN}$ such that $\textbf A_{MN}[v]_M = [v]_N$
The bases are not given. The only information given is $\hat u_M$, $\hat u_N$, $\hat v_M$ and $\hat v_N$. I believe it is possible to construct said transformation matrix, but I don't know how to do it.
Specifically, I am concerned with the theoretical problem of attitude determination onboard a satellite. Given satellite-measured magnetic field and sun direction vectors, and independent reference frame magnetic field and sun-direction vector, how do you construct $\textbf A_{MN}$ as accurately as possible to evaluate the satellite $\textbf Z^+$ vector in the independent reference frame?