IMC 2009 · Problem 5

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S \subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$.

Say that a vector subspace $T \subseteq \mathbb{M}$ is a covering matrix space if

$$\bigcup_{A \in T, \, A \ne 0} \ker A = \mathbb{R}^p.$$

Such a $T$ is minimal if it does not contain a proper vector subspace $S \subset T$ which is also a covering matrix space.

(a) Let $T$ be a minimal covering matrix space and let $n = \dim T$. Prove that

$$\delta(T) \le \binom{n}{2}.$$

(b) Prove that for every positive integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T = n$ and $\delta(T) = \binom{n}{2}$.