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- For all $u,v \in \ZZ^2$, there exist $x,y \in \ZZ^2$, such that:
$$ax + by = u,$$ $$cx + dy = v.$$
By State/Systems of two linear Diophantine equations, this is equivalent to the following condition:
- $ad - bc = \pm 1$.
From this equivalent condition, it is clear that the notion of lax basis is well-defined; it depends only upon the unordered pair of lax vectors $[a,b]$ and $[c,d]$, and not upon a choice of representatives $\pm (a,b)$ and $\pm (c,d)$ in $\ZZ^2$.
The following statements and definitions logically rely on the material of this page: Def/Domain topograph, Def/Lax superbasis, State/Arithmetic progression rule for binary quadratic forms, State/Every lax basis is contained in exactly two lax superbases, State/Every lax vector in a lax basis is primitive, and State/Every primitive lax vector belongs to a lax basis
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Binary quadratic forms