Linear representation of matroids
The axioms of matroid theory are abstracted from the properties of linear dependence. Thus, a natural question is whether every matroid can be represented by a set of vectors with coordinates over some field k so that abstract dependence in the matroid coincides with linear dependence of the vectors. If such a set of vectors exists, the matroid is said to be representable over the field k. This question was first considered by Whitney in [I.1], §16. There he gave the Fano plane as an example of a matroid representable only over a field of characteristic 2, and hence, not representable over the real or complex numbers. The question whether there exists a matroid not representable over any field is answered in Mac Lane’s paper, “Some interpretation of abstract linear dependence in terms of projective geometry,” the first paper reprinted in this chapter.
KeywordsElementary Chain Connected Line Connected Plane MATROID Representation Fano Plane
Unable to display preview. Download preview PDF.
- Higgs, D.A.: Geometry, Lecture notes, Univ. Waterloo, Waterloo, Ont., 1967.Google Scholar
- Ingleton, A.W.: Representations of matroids, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 149–167, Academic Press, London, 1971.Google Scholar
- Mason, J.H.: Matroids as the study of geometric configurations, Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht, 1977.Google Scholar
- Reid, R.: Obstructions to representations of combinatorial geometries, unpublished; appeared as an appendix to T. Brylawski and D.G. Kelly, Matroids and Combinatorial Geometries, Univ. North Carolina Press, Chapel Hill, N.C., 1980.Google Scholar
- Vámos, P.: Linearity of matroids over division rings (Notes by G. Roulet), Möbius Algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 170–174, Univ. Waterloo, Waterloo, Ont., 1971.Google Scholar
- Vámos, P.: A necessary and sufficient condition for a matroid to be linear, Möbius Algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 162–169, Univ. Waterloo, Waterloo, Ont., 1971.Google Scholar
- White, N.L.: Coordinatization of combinatorial geometries, Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1970), pp. 484–486, Univ. North Carolina, Chapel Hill, N.C., 1970.Google Scholar
- 3.F. Harary and D. J. A. Welsh, “Matroids versus Graphs,” Lecture Notes in Mathematics, Vol. 110, pp. 155–170, Springer, Berlin, 1969.Google Scholar
- 4.A. W. Ingleton, Representation of matroids, in “Combinatorial Mathematics and Its Applications,” pp. 149–167, Academic Press, New York, 1971.Google Scholar
- 9.W. T. Tutte, “Introduction to the Theory of Matroids,” RAND Corp. Report R-448-PR, 1966.Google Scholar
- 13.T. H. Brylawski and T. D. Lucas, Uniquely representable combinatorial geometries, in “Proc. Internat. Colloq. Combinatorial Theory, Rome, Italy, 1975,” pp. 83–104.Google Scholar