  In this paper, we introduce the notions of matching matrices in groups and
vector spaces, which lead to some necessary conditions for existence of acyclic
matching in abelian groups and its linear analogue. We also study the linear
local matching property in field extensions to find a dimension criteria for
linear locally matchable bases. Moreover, we define the weakly locally
matchable subspaces and we investigate their relations with matchable
subspaces. We provide an upper bound for the dimension of primitive subspaces
in a separable field extension. We employ MATLAB coding to investigate the
existence of acyclic matchings in finite cyclic groups. Finally, a possible
research problem on matchings in n-groups is presented. Our tools in this paper
mix combinatorics and linear algebra.
