Find conditions on a, b, c, d
Find conditions on a, b, c, d $\displaystyle \in \mathbb{Z}$ for {(a, b), (c, d)} to be a basis for $\displaystyle \mathbb{Z}\times\mathbb{Z}$. [Hint: Solve x(a, b) + y(c, d) = (e, f) in $\displaystyle \mathbb{R}$, and see when the x and y lie in $\displaystyle \mathbb{Z}$.]
My solution:
x(a, b) + y(c, d) = (e, f)
ax + cy = e
bx + dy = f
x = (ed - fc)/(ad - bc)
y = (af - be)/(ad - bc)
Since x and y $\displaystyle \in \mathbb{Z}$, $\displaystyle ad - bc =\pm 1$ is the required condition.
Am I right?
Re: Find conditions on a, b, c, d
