  In the theory of classical matroids, there are several known equivalent
axiomatic systems that define a matroid, which are described as matroid
cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play
the role of the subsets in the classical theory. In this article we establish
cryptomorphisms of q-matroids. In doing so we highlight the difference between
classical theory and its q-analogue. We introduce a comprehensive set of
q-matroid axiom systems and show cryptomorphisms between them and existing
axiom systems of a q-matroid. These axioms are described as the rank, closure,
basis, independence, dependence, circuit, hyperplane, flat, open space,
spanning space, non-spanning space, and bi-colouring axioms.
