Abstract
A problem concerning the dimensions of the intersections of a subspace in the direct sum of a finite number of the finite-dimensional vector spaces with the pairwise sums of direct summands, provided that the subspace intersection with these direct summands is zero has been discussed. The problem is naturally divided into two ones, namely, the existence and representability of the corresponding matroid. Necessary and sufficient conditions of the existence of a matroid with the specified ranks of some subsets of the base set have been given. Using these conditions, necessary conditions of the existence of a matroid with a base set composed of a finite series of pairwise disjoint sets of the full rank and the given ranks of their pairwise unions have been presented. A simple graphical representation of the latter conditions has also been considered. These conditions are also necessary for the subspace to exist. At the end of the paper, a conjecture that these conditions are sufficient has also been stated.
Similar content being viewed by others
References
J. G. Oxley, “What is a matroid?” Cubo 5, 179–218 (2003).
M. M. Shikare and B. N. Waphare, Combinatorial Optimization (Narosa, New Delhi, 2004).
4ti2 team. 4ti2–A software package for algebraic, geometric and combinatorial problems on linear spaces. http://www.4ti2.de.
J. Ellson, E. Gansner, L. Koutsofios, S. North, and G. Woodhull, “Graphviz–open source graph drawing tool,” in Graph Drawing, in Ser. Lecture Notes in Computer Scienc, Vol. 2265 (Springer-Verlag, Berlin, 2002), pp. 483–484.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.A. Lebedinskaya, D.M. Lebedinskii, 2016, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2016, No. 2, pp. 204–209.
About this article
Cite this article
Lebedinskaya, N.A., Lebedinskii, D.M. On the possible dimensions of subspace intersections. Vestnik St.Petersb. Univ.Math. 49, 115–118 (2016). https://doi.org/10.3103/S1063454116020096
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1063454116020096