Say I have a vector basis $A$ assume 2D for now. If I have an expression of a basis B in terms of A as 2 orthonormal vectors $v_1, V_2$v it's trivial to see how the matrix $[v_1, v_2]$ acts on a vector expressed in $A$.
$v_1 \cdot \vec{u}$ yields the component of $\vec u$ in the direction of $v_1$ and $v_2 \cdot \vec{u}$ the component in the direction of $v_2$. So the result is immediate.
Now let's say the vectors are orthogonal but not orthonormal.
$v_1 \cdot \vec{u} - \|v_1\|\|u\|\cos(\theta)$
This is not quite the component of $u$ relative to $v_1$ for example if $v_1 = [2, 0]$ and $u = [4, 0]$ then $v_1 \cdot u = 8$ but if we wanted to express $u$ in terms of our selected basis it should be $u = 2 v_1 + 0 v_2$.
Thus the observation for orthonormal vectors does not generalize to orthogonal vectors. But it does seem to be the case that $B = [v_1, v_2]$ eats vectors expressed in $B$ and spits out their representation in $A$.
What if the vectors are not even orthogonal.
Is it still the case that $B$ eats vectors expressed in its cooridnate system and spits out vectors expressed in the coordinate system of $A$?