Question

Hello everyone,

In ${\mathbb R}^n$, we say that a linear subspace is \emph{rational} if it admits a basis in ${\mathbb Q}^n$ (or equivalently in ${\mathbb Z}^n$). It means that $E\cap {\mathbb Z}^n$ is a submodule of {\mathbb Z}^n of rank equal to the dimension of the given subspace. By convention, ${0}$ is rational.

Now, if $E$ is any linear subspace of ${\mathbb R}^n$, we denote $b(E)$ (resp. $c(E)$ ) the dimension of the biggest rational subspace of ${\mathbb R}^n$ contained in $E$ (resp. the smallest rational subspace of ${\mathbb R}^n$ containing $E$).

My questions: 1) How can one compute $b(E)$ and $c(E)$? I am looking mainly for an algorithm (if it is possible). My entries for the algorithm are either a basis of $E$, or a system of equations (or any equivalent characterization).

2) It is clear that $0\leq b(E)\leq dim(E)\leq c(E)\leq n$. From an article, every case is possible. How one can construct such examples of different cases? I have some ideas but I am stuck in verifying because of the first question.

Thanks in advance.

(Maybe the question is simple for specialists. In this case, I would be grateful if someone could provide me a reference).

Answer 1

Given a matrix $M$ whose kernel is $E$, let ${\alpha_1, \ldots, \alpha_k}$ be a basis for the linear span over $\mathbb Q$ of the entries of $M$. Then $M = \sum_{j=1}^k \alpha_j M_k$ where $M_j$ are matrices with rational entries. Let $N$ be the matrix obtained by stacking the $M_j$ vertically.
Any member of $E \cap {\mathbb Q}^n$ is in the kernel of $N$, while the kernel of $N$ has a basis consisting of vectors in ${\mathbb Q}^n$. Thus $b(E)$ is the dimension of the kernel of $N$.

EDIT: In the other direction, consider the annihilator $E^\perp = \{y \in {\mathbb R}^n: y \cdot x = 0 \ \text{for all}\ x \in E\}$. Note that the annihilator of a rational subspace is rational (a basis in ${\mathbb Q}^n$ can be obtained using Gaussian elimination). It follows that the smallest rational subspace containing $E$ is the annihilator of the largest rational subspace contained in $E^\perp$. Thus $c(E) = n - b(E^\perp)$.