  In the first part of this paper we develop some theorems in linear algebra
applicable to information theory when all random variables involved are linear
functions of the individual bits of a source of independent bits.
  We say that a collection of subspaces of a vector space are "coordinated" if
the vector space has a basis such that each subspace is spanned by its
intersection with the basis. We measure the failure of a collection of
subspaces to be coordinated by an invariant that we call the "discoordination"
of the family. We develop some foundational results regarding discoordination.
In particular, these results give a number of new formulas involving three
subspaces of a vector space.
  We then apply a number of our results, along with a method of Tian to obtain
some new lower bounds in a special case of the basic coded caching problem. In
terms of the usual notation for these problems, we show that for $N=3$
documents and $K=3$ caches, we have $6M+5R\ge 11$ for a scheme that achieves
the memory-rate pair $(M,R)$, assuming the scheme is linear. We also give a new
caching scheme for $N=K=3$ that achieves the pair $(M,R) = (1/2,5/3)$.
