Prove that the elements $b_{1} = (x_{1}, y_{1})$ and $b_2 = (x_{2}, y_{2})$ form a basis of $\mathbb{Z} \oplus \mathbb{Z}$ if, and only if, $x_{1}y_{2} - x_2y_1 = ± 1$
I was able to show that these vectors are linearly independent because the determinant between them is by hypothesis $\pm 1$. But I could not show that they generate all $\mathbb {Z} \oplus \mathbb {Z} $. How to proceed?
Whenever a matrix $A$ has coefficients in a commutative ring $R$ and its determinant is a unit in $R$, then $A$ has an inverse with coefficients in $R$. We have that $$\det \begin{pmatrix} x_1 & x_2 \\ y_1 & y_2 \end{pmatrix} = x_1y_2 - x_2y_1.$$ Applying this to your setting, your assumption reads that $A$ is invertible with inverse $A^{-1}$. Using this, let $v \in \mathbb Z^2$ be arbitrary. Consider $w = A^{-1}v$. Then $Aw = v$, so $v$ is a linear combinations of the columns of $A$.
