If I am given some lattice defined as, say
$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$
and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that $v$ and $w$ form a basis for $L$.
I'm a bit stuck, but I can see how this would be accomplished in a lattice where $z_1=1$ and $z_2=i$ if I was given $v=1+i$: inspecting the lattice I could choose $w=i$ (or $w=1$) and still cover all the lattice points (diagonal lines of slope $1$ along all the lattice points).
If $v=2+i$, I can see how $w=1+i$ would work, where along each row of lattice "squares" you could could go across $2$, up $1$ ($v=2+i$), then up $1$, across $1$ ($w=1+i$), then subtract a $v$, so you are now at
$(2+i)+(1+i)-(2+i)=1+i=w$
then add a $w$ so you are at $2+2i$, then subtract a $v$ so you are at
$(2+2i)-(2+i)=i$
and continue ad nauseum.
My questions are:
- In general, are we only guaranteed a single choice of $w$?
- How can I use the fact that there are always $s,t$ where $sa+tb=1$ to help find a a $w$?
Thanks in advance.