1. Introduction. The purpose of this paper is to prove that all y-matroids on a finite set E have vector representations over any field with at least 2 |£| elements. We give a new proof and a generalization of a recent result by J. H. Mason on matroids induced in a directed graph.A matroid (or finite independence space) on a finite set £ is a pair (E, M), where M is a collection of subsets of E with the properties (ii) XeM and Y ^X implies that YeM, (iii) X, Y e M and \Y\ = |X| + 1 imply that Xu{e}eM for some element e e Y -X . fThe sets which belong to M are called the independent subsets of E. Subsets of E, which are not independent, are called dependent. An element e e E such thatA representation of a matroid (£, M) over a field F is a function / which maps E into a vector space over F such that the image/(X) of the elements in any independent set X is a set of \X\ vectors, which are linearly independent over F. Loops are mapped on the zero vector.Let stf = (A t : iel) be a finite family of subsets of a finite set E. A partial transversal of «s/ is a subset X c £ for which there is an injective mapping (j>: X -> J such that xe A^X) when x e l . J. Edmonds and D. R. Fulkerson proved in [2] that the partial transversals of stf and 0 are the independent sets of a matroid on E called the transversal matroid of J^. L. Mirsky and H. Perfect proved in [5] that any transversal matroid has a vector representation over some transcendental extension of the rational field, and M. J. Piff and D. J. A. Welsh proved in [8] that there is a representation over any sufficiently large field. Transversal matroids are special instances of the y-matroids.A directed graph consists of two sets; one set of nodes N and a set of edges, which is a subset of the cartesian product NxN. We do not admit loops (g, g). Let F be a directed graph with two distinguished subsets of nodes X and Y, which need not

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Given an integer k ≤ 2 and a finite set M of rational integers. Let vi (i = 1, 2, …, n) be m-dimensional (column-)vectors with all components from M and such that the kn sums1.1are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.

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