0
$\begingroup$

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

  • 4
    Could you provide some context for this question? Is it a homework problem?2010-08-11
  • 3
    Moreover your title is super general, you should make it more specific.2010-08-11
  • 0
    @BBischof: Please help me in renaming the title. Like what could i name it.2010-08-12
  • 0
    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.2010-08-15

1 Answers 1

5

Problem 5

  • 0
    Nice. How in world did you find that?2010-08-12