MathDB

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 \emph{covering matrix space} if

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

Such a

T

is minimal if it doesn't contain a proper vector subspace

S\subset T

such that

S

is also a covering matrix space.
(a) (8 points) Let

T

be a minimal covering matrix space and let

n=\dim (T)

Prove that

\delta(T)\le \dbinom{n}{2}

(b) (2 points) Prove that for every 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)=\dbinom{n}{2}

Problem Statement