For the two bases, $A$ and $B$ whose columns represent the basis vectors respectively, I was asked to find the transformation matrix from $A$ to $B$
$$ A= \begin{Bmatrix} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 1 & -1 & 3 \\ \end{Bmatrix} $$
$$ B= \begin{Bmatrix} -3 & 1 & 5 \\ 2 & -1 & 4 \\ -2 & -1 & 9 \\ \end{Bmatrix} $$
I began by comparing the the basis vectors in $A$ to those in $B$ as per the definition of what the change of basis matrix is - it contains the coordinates of the new basis wrt to the old basis e.g.
$$ \left(\begin{matrix} 1 \\ 2 \\ 1 \\ \end{matrix}\right) = x_1\left(\begin{matrix} -3 \\ 2 \\ -3 \\ \end{matrix}\right) + x_2\left(\begin{matrix} 1 \\ -1 \\ -1 \\ \end{matrix}\right) + x_3\left(\begin{matrix} 5 \\ 4 \\ 9 \\ \end{matrix}\right) $$
And did so for all other columnns. This does seem like a tedious way to do things. Would I be right in thinking that I could identify the values of x, which are the coordinate representations of $A$ wrt $B$ by through the following relationship:
$A = XB$ and therefore, assuming $B$ is non-singular I could find the coordinates X by computing:
$AB^{-1} = X$ ?
If this is invalid could some explain why and perhaps suggest a quicker solution to such an equation?
The equation $ \left(\begin{matrix} 1 \\ 2 \\ 1 \\ \end{matrix}\right) = x_{11}\left(\begin{matrix} -3 \\ 2 \\ -3 \\ \end{matrix}\right) + x_{21}\left(\begin{matrix} 1 \\ -1 \\ -1 \\ \end{matrix}\right) + x_{31}\left(\begin{matrix} 5 \\ 4 \\ 9 \\ \end{matrix}\right) $ can be written as $$ \left(\begin{matrix} 1 \\ 2 \\ 1 \\ \end{matrix}\right) = \left(\begin{matrix} -3&1&5 \\ 2&-1&4 \\ -3&-1&9 \\ \end{matrix}\right)\left(\begin{matrix} x_{11} \\ x_{21} \\ x_{31} \\ \end{matrix}\right) $$ Do you see how you $X$ matrix should be on the right and not the left?