<s>[INST] Computing the Change of Basis Matrix between two vector spaces 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?
 [/INST] 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?
</s>