# Math Help - Find conditions on a, b, c, d

1. ## Find conditions on a, b, c, d

Find conditions on a, b, c, d $\in \mathbb{Z}$ for {(a, b), (c, d)} to be a basis for $\mathbb{Z}\times\mathbb{Z}$. [Hint: Solve x(a, b) + y(c, d) = (e, f) in $\mathbb{R}$, and see when the x and y lie in $\mathbb{Z}$.]

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 $\in \mathbb{Z}$, $ad - bc =\pm 1$ is the required condition.

Am I right?