  In this paper, we formulate and prove linear analogues of results concerning
matchings in groups. A matching in a group G is a bijection f between two
finite subsets A,B of G with the property, motivated by old questions on
symmetric tensors, that the product af(a)does not belong to A for all a \in A.
Necessary and sufficient conditions on G, ensuring the existence of matchings
under appropriate hypotheses, are known. Here we consider a similar question in
a linear setting. Given a skew field extension K \subset L, where K commutative
and central in L, we introduce analogous notions of matchings between
finite-dimensional K-subspaces A,B of L, and obtain existence criteria similar
to those in the group setting. Our tools mix additive number theory,
combinatorics and algebra.
