  Seymour's decomposition theorem for regular matroids is a fundamental result
with a number of combinatorial and algorithmic applications. In this work we
demonstrate how this theorem can be used in the design of parameterized
algorithms on regular matroids. We consider the problem of covering a set of
vectors of a given finite dimensional linear space (vector space) by a subspace
generated by a set of vectors of minimum size. Specifically, in the Space Cover
problem, we are given a matrix M and a subset of its columns T; the task is to
find a minimum set F of columns of M disjoint with T such that that the linear
span of F contains all vectors of T. For graphic matroids this problem is
essentially Stainer Forest and for cographic matroids this is a generalization
of Multiway Cut. Our main result is the algorithm with running time
2^{O(k)}||M|| ^{O(1)} solving Space Cover in the case when M is a totally
unimodular matrix over rationals, where k is the size of F. In other words, we
show that on regular matroids the problem is fixed-parameter tractable
parameterized by the rank of the covering subspace.
