# Dimension of Vector space

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 $$\delta(T) \leq { n \choose 2 }$$

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Could you provide some context for this question? Is it a homework problem? – Pete L. Clark Aug 11 '10 at 20:32
Moreover your title is super general, you should make it more specific. – BBischof Aug 11 '10 at 20:33
@BBischof: Please help me in renaming the title. Like what could i name it. – anonymous Aug 12 '10 at 4:03
Hehe, I don't see a obviously good name, but maybe something like "Vector Space Dimension Inequality for Covering Matrix Spaces". That is the only one I can come up with at this point. – BBischof Aug 15 '10 at 22:23