This problem is taken from International Mathematics Competition for University Students 2009 (IMC 2009), Day 2, Problem 5.
Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S \subset \mathbb{M}$, denote $\delta(S)$ the dimesion of the vector space generated by all columns of all matrices in $S$. Say that a vector space $T \subset \mathbb{M}$ is a covering space, if $$ \bigcup\limits_{A \in T, A \neq 0} ker(A) = \mathbb{R}^{p}$$
Such a $T$ is minimal if it does not contain a proper vector subspace $S \subset T$ which is also a covering matrix space. Let $T$ be a minimal covering matrix space and let $n= \text{dim}(T)$. Prove that $$\delta(T) \leq { n \choose 2 } $$