Let $a,b,c$ be integers and let $[a,b,c]$ denote the mapping

Let $G$ be the group of $2\times 2$ matrices such that $\det{A}=\pm 1\;\forall\;A\in G$. The substitution

leads to

where

So we define

to be the binary quadratic form with coefficients $a^{{{}^{{\prime}}}},b^{{{}^{{\prime}}}},c^{{{}^{{\prime}}}}$ of $x^{2},xy,y^{2}$, respectively as in (1). Putting in $A=\begin{matrix}1&0\\ 0&1\end{matrix}$ we have $[a,b,c]\ast A=[a,b,c]$ for any binary quadratic form $[a,b,c]$. Now let $B$ be another matrix in $G$. We must show that

Set $[a,b,c]\ast(AB):=[a^{{{}^{{\prime\prime}}}},b^{{{}^{{\prime\prime}}}},c^{{{}^{{\prime\prime}}}}]$. So we have

as desired. For the coefficient $b^{{{}^{{\prime\prime}}}}$ we get

and by evaluating the factors of $b_{{11}}b_{{12}},b_{{21}}b_{{22}}$, and $b_{{11}}b_{{22}}+b_{{21}}b_{{12}}$, it can be checked that

This shows that

and therefore $[a,b,c]\ast(AB)=([a,b,c]\ast A)\ast B$. Thus, (1) defines an action of $G$ on the set of (integer) binary quadratic forms. Furthermore, the discriminant of each quadratic form in the orbit of $[a,b,c]$ under $G$ is $b^{2}-4ac$.