<s>[INST] Linear Algebra Working with Linear Transformations Let $v_1=[-3;-1]$ and $v_2= [-2;-1]$
Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation satisfying:
$T(v_1)=[15;-6]$ and $T(v_2)=[11;-3]$
Find the image of an arbitrary vector $[x;y]$
 [/INST] Note sure if (homework) yet. So hint:
Let 
$$
T = 
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
$$
We can re-interpret the given $T(v_1)$ and $T(v_2)$ as:
$$
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\begin{pmatrix}
-3 \\ -1
\end{pmatrix}
=
\begin{pmatrix}
15 \\ -6
\end{pmatrix} ,
\\
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\begin{pmatrix}
-2 \\ -1
\end{pmatrix}
=
\begin{pmatrix}
11 \\ -3
\end{pmatrix} 
$$
Or more succinctly as,
$$
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\begin{pmatrix}
-3 & -2 \\ -1 & -1
\end{pmatrix}
=
\begin{pmatrix}
15 & 11 \\ -6 & -3
\end{pmatrix} 
\tag{1}
$$
Can you take it from here?
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