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

$\endgroup$
4
  • 4
    $\begingroup$ Could you provide some context for this question? Is it a homework problem? $\endgroup$ Aug 11, 2010 at 20:32
  • 3
    $\begingroup$ Moreover your title is super general, you should make it more specific. $\endgroup$
    – BBischof
    Aug 11, 2010 at 20:33
  • $\begingroup$ @BBischof: Please help me in renaming the title. Like what could i name it. $\endgroup$
    – anonymous
    Aug 12, 2010 at 4:03
  • $\begingroup$ 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. $\endgroup$
    – BBischof
    Aug 15, 2010 at 22:23

1 Answer 1

5
$\begingroup$

Problem 5

$\endgroup$
1
  • $\begingroup$ Nice. How in world did you find that? $\endgroup$
    – Mykie
    Aug 12, 2010 at 16:40

You must log in to answer this question.