<s>[INST] Which one is the correct way to compute the change of basis matrix? I was looking for a way to compute the change of basis matrix (given the old basis and the new basis), but I found two methods that lead to different results, and I can't understand which one is correct.    
Method 1:
$$ 
B = \left\{
\begin{pmatrix}
1 \\ 2 
\end{pmatrix} , 
\begin{pmatrix}
3 \\ 4 
\end{pmatrix} 
\right\} ~ , ~
 D= \left\{
\begin{pmatrix}
1 \\ 4 
\end{pmatrix} , 
\begin{pmatrix}
2 \\ 3 
\end{pmatrix} \right\} 
$$
The vectors in $D$ are expressed as a linear combination of the ones in $B$, and then the coefficients are used to construct the change of basis matrix $S$, i.e.:
$$
\begin{pmatrix}1 \\ 4 \end{pmatrix} = s_{11}\cdot \begin{pmatrix}1 \\ 2 \end{pmatrix} + s_{12} \cdot \begin{pmatrix}3 \\ 4 \end{pmatrix} \ \ \Rightarrow s_{11}=4 \ \ \text{and}  \ \ s_{12}=-1  \\
\begin{pmatrix}2 \\ 3 \end{pmatrix} = s_{21}\cdot \begin{pmatrix}1 \\ 2 \end{pmatrix} + s_{22} \cdot \begin{pmatrix}3 \\ 4 \end{pmatrix} \Rightarrow s_{21}=\frac{1}{2} \ \ \text{and}  \ \ s_{22}=\frac{1}{2}  \\
\Rightarrow S =\begin{pmatrix}4 & -1 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}
$$ 
Method 2:
$$\begin{pmatrix}s_{11} & s_{12} \\ s_{21} & s_{22} \end{pmatrix} \cdot \begin{pmatrix}1 & 3 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix}1 & 2 \\ 4 & 3 \end{pmatrix} \\
\Rightarrow s_{11}=0 \ \ ;\ \ s_{12}= \frac{1}{2} \ \ ; \ \ s_{21}= -5 \ \ ; \ \ s_{22}=\frac{9}{2}\\
\Rightarrow S =\begin{pmatrix}0 & \frac{1}{2} \\ -5 & \frac{9}{2} \end{pmatrix}
$$
 [/INST] The method 2) is correct. It is the same as to find a matrix
$$
S=\begin{pmatrix}s_{11} & s_{12} \\ s_{21} & s_{22} \end{pmatrix} 
$$
such that 
$$
S\begin{pmatrix}1 \\2 \end{pmatrix}=\begin{pmatrix}1 \\4 \end{pmatrix}
$$
and
$$
S\begin{pmatrix}3 \\4 \end{pmatrix}=\begin{pmatrix}2 \\3 \end{pmatrix}
$$
that means exactly to transform the vectors of the first basis to the vectors of the second basis.
I don't see how the method 1) comes from...but it is wrong.
</s>