I understand how we can find the transformation matrix $D$ with respect to another basis $B$, by using a transformation matrix that we already know, say $A$: $$D = C^{-1}\cdot A\cdot C$$
Where $C$ is the change of basis for the basis $B$ and $C^{-1}$ is its inverse.
Now, this is fine, but there's a concrete problem that gives different results if we first multiply $A.C$ rather than $C^{-1} \cdot A$
I am watching a video, and specifically at min 10.25, the professor starts by first multiplying $A.C$ and not $C^{-1}.A$, which is what I did, obtaining at the end a different $D$.
My questions are:
How the order that we use to multiply the matrices can change the resulting $D$ transformation matrix?
Which version is correct:
- Multiply first $C^{-1}$ by $A$ or multiply first $A$ by $C$, and why?
I have used Maxima to solve this problem and apparently it has done what I did: it has multiplied first $C^{-1}$ by $A$...obtaining at the end the same matrix I obtained, as you can see from the picture:
Just for completeness:
$$B = \{ \left[ \begin{matrix} 1 \\ 2 \end{matrix}\right], \left[ \begin{matrix} 2 \\ 1 \end{matrix}\right] \}$$
$$A = \left[ \begin{matrix} 3 & -2 \\ 2 & -3 \end{matrix}\right]$$