Let $A[a,b,c]$ and $B[d,e,f]$ be two non-standard bases. I have to find the $3\times3$ matrix that will convert a vector defined in terms of $A$ to $B$.
My solution is:
Let's assume a standard basis $O[e_1,e_2,e_3]$.
Now, the rotation matrix $R(A\to O)$ to change basis from $A$ to $O$ is:
$ \begin{bmatrix} ax & bx & cx \\ ay & by & cy \\ az & bz & cz \end{bmatrix} $
and $R(O\to B)$ is
$ \begin{bmatrix} ax & ay & az \\ bx & by & bz \\ cx & cy & cz \end{bmatrix}. $
So, $R(A \to B)=R(A \to O)\cdot R(O\to B)$.
Is this the correct solution?
Thanks a lot.