# Finding an integer orthogonal basis

Say I have some non-orthogonal basis of some vector space that only have integer elements. Is it possible to find an orthogonal basis consisting of basis vectors with integer elements?

For example, an orthogonal basis of the span of $[-3,0,2]$ and $[3,5,0]$ could be given by $[-57,0,38]$ and $[12,65,18]$.
More formally, if $a_1,\ldots,a_n\in\mathbb{Q}^m$ then the $k$th vector of the orthogonal basis generated by the Gram-Schmidt orthogonalization is $$q_k=a_k-\sum_{j=1}^{k-1}\frac{\langle q_{j},a_k \rangle}{\langle q_{j},q_j\rangle}q_j,$$ where $\langle\cdot,\cdot\rangle$ is the Euclidean inner product. You can use the induction to show that the orthogonalization coefficients are rational and hence each vector $q_k$ has the form $q_k=[\frac{r_1}{s_1},\ldots,\frac{r_n}{s_n}]$, where $r_i\in\mathbb{Z}$ and $s_i\in\mathbb{Z}^+$. Setting $\tilde{q}_k=\mathrm{lcm}(s_1,\ldots,s_n)\times q_k$ then provides the orthogonal basis $\tilde{q}_1,\ldots,\tilde{q}_k\in\mathbb{Z}^n$ of $\mathrm{span}(a_1,\ldots,a_n)$.