Determine matrix of linear map Linear map is given through:
$\phi\begin{pmatrix} 3 \\ -2 \end{pmatrix} =\begin{pmatrix} -3 \\ -14 \end{pmatrix} $
$\phi\begin{pmatrix} 3 \\ 0 \end{pmatrix} =\begin{pmatrix} -9 \\ -6 \end{pmatrix}$
Determine matrix $A$ linear map.
Here I have solution but I dont understand how to get it.
$A=\begin{pmatrix} -3 & -3 \\ -2 & 4 \end{pmatrix}  $

One possibility is "brute force" solution: you know $A\pmatrix{3 \\ -2} = \pmatrix{-3 \\ -14}, A\pmatrix{3 \\ 0} = \pmatrix{-9 \\ -6}$, that's four equations for four unknown elements, solvable in a number of ways. This approach is inelegant and quickly becomes hard for bigger dimensionality, but is perfectly doable.
The other is that you know matrix of $\phi$ in basis $b' = (\pmatrix{3 \\ -2}, \pmatrix{3 \\ 0})$ (it is $A' = \pmatrix{-3 & -9 \\ -14 & -6}$) and you want to learn what matrix it has in basis $b_0 = (\pmatrix{1 \\ 0}, \pmatrix{0 \\ 1})$. That would be $A'*\pmatrix{3 & 3 \\ -2 & 0}^{-1} = A'*\pmatrix{0 & -{1 \over 2} \\ 1 \over 3 & 1 \over 2} = \pmatrix{-3 & -3 \\ -2 & 4}$.
