<s>[INST] Find $2\times 2$ matrix of linear transformation $T$ Given:
\[
T\left(\begin{bmatrix}-1 \\ -2\end{bmatrix}\right) = \begin{bmatrix}17 \\ 11\end{bmatrix} \text{  and  } T\left(\begin{bmatrix}2 \\ 3\end{bmatrix}\right) = \begin{bmatrix}-30 \\ -17\end{bmatrix}
\]
Find a matrix such that:
\[ T(v) = \begin{bmatrix}? & ? \\ ? & ? \end{bmatrix}\cdot v \]
I have no idea where to start... Any help will be much appreciate it.
 [/INST] Let's write $$\begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ for the matrix we want to find. Then the two given equations read 
\begin{align*}
  -a - 2b &= 17\\
  -c - 2d &= 11\\[3mm]
  2a + 3b &= -30\\
  2c + 3d &= -17
\end{align*}
We have to solve this system, so let's first look at the equations for $a$ and $b$
\begin{align*}
  -a - 2b &= 17\\
  2a + 3b &= -30
\end{align*}
Adding the first two times to the second gives
\begin{align*}
  -a - 2b &= 17\\
       -b &= 4
\end{align*}
Hence $b = -4$ and $a = -9$. The second system reads
\begin{align*}
  -c - 2d &= 11\\
  2c + 3d &= -17
\end{align*}
Adding again, we have
\begin{align*}
  -c - 2d &= 11\\
       -d &= 5
\end{align*}
Hence $d = -5$ and $c = -1$.
So $$ \begin{bmatrix} -9 & -4 \\ -1 & -5\end{bmatrix} $$ is the matrix we looked for.
</s>