<s>[INST] Find the change of basis matrix P from S to S'. Consider the following bases of $\mathbb{R}^2$:
$$S=\left\{\begin{pmatrix} 
1\\
-2 
\end{pmatrix},\begin{pmatrix} 
3 \\
-4 
\end{pmatrix}\right\}$$
$$S'=\left\{\begin{pmatrix} 
1\\
3
\end{pmatrix},\begin{pmatrix} 
3 \\
8 
\end{pmatrix}\right\}$$
Find the change of basis matrix P from S to S'.
So I followed a guide online which said to express S in terms of S':
$$\begin{pmatrix} 
1\\
-2 
\end{pmatrix}= \begin{pmatrix} 
1\\
3
\end{pmatrix}a+\begin{pmatrix} 
3 \\
8 
\end{pmatrix}b$$
$$\begin{pmatrix} 
3\\
-4 
\end{pmatrix}= \begin{pmatrix} 
1\\
3
\end{pmatrix}c+\begin{pmatrix} 
3 \\
8 
\end{pmatrix}d$$
So I got $a=-14,b=5,c=-36,d=13$ so would the answer simply be:
$$\begin{pmatrix} 
-14 & 5\\
-36 & 13
\end{pmatrix}$$
or am I missing a step?
 [/INST] With your notation, I think your final matrix should be 
$$\begin{pmatrix} a&c\\b&d \end{pmatrix}$$
</s>