<s>[INST] Show that the vectors $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are linearly independent... Show that the vectors $v_1 = \begin{pmatrix}
1 \\
2
\end{pmatrix}$ and $v_2 = \begin{pmatrix}
2 \\
3
\end{pmatrix}$ are linearly independent, and find the unique coefficients $x$ with
\begin{equation*}
x_1v_1+x_2v_2 = [v_1, v_2]x = \begin{pmatrix}
a \\
b
\end{pmatrix}.
\end{equation*}
To show that the vectors are independent we just row reduce the matrix containing the two vectors to make sure we get the identity matrix right.
My question is how do we find the unique constants with that equation shown?
Thank you.
 [/INST] Another way is just to note that two vectors are linearly independent as long as they're not multiples of each other. 
The hint by @Integrate This leads to $\begin {pmatrix}x_1\\x_2\end{pmatrix}=\begin{pmatrix}-3&2\\2&-1\end{pmatrix}\begin {pmatrix}a\\b\end{pmatrix}$, after inverting the $2×2$ matrix.
</s>