# orthogonal vectors in the row space of set inclusion matrices

Consider the set-inclusion matrix M of size $\binom{n}{t}\times \binom{n}{k}$ whose entries are given by $M(T,K)=[T\subseteq K]$ (where $[\cdot]$ is $1$ is $\cdot$ is satisfied and $0$ otherwise) for all $T\subseteq [n],K\subseteq [n]$ and $|T|=t,|K|=k$ (where $t\leq k$).

Is there any reference that looks at the vectors satisfying $Mx=0$? I want to understand Kernel of this matrix. Thanks for any pointer

• The matrix $M^TM$ is symmetric semi positive definite, and has the same rank and kernel of $M$, so probably it is better to study the first – Exodd Feb 24 '18 at 18:04