Applied mathematics

Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo $2$

E. A. Kalinina, G. M. Khitrov

St. Petersburg State University, 199034, St. Petersburg, Russian Federation

Abstract: In the paper, vector spaces over residue field modulo $2$ are considered. These vector spaces are of considerable interest because they are widely used in ordinary graphs theory, theory of coding and others areas, especially in modular systems investigation. Vector spaces over $\mathrm{GF(2)}$ have some features, for example, examination of linear dependence and independence for the set of vectors is simplified. The concept of $1$-dependence for the set of vectors is embedded. This concept is used to study vector subspaces and their orthogonal complements and to solve systems of linear equations. The connection between fundamental system of solutions of some simultaneous linear equations and decomposition of corresponding vector system into minimal $1$-dependent subsystems is considered. The necessary and sufficient conditions for the existence of nontrivial intersection of the vector subspace and its orthogonal complement are proven. Bibliogr. 10.

Keywords: vector space, residue field modulo 2.

UDC: 512.642

Received: October 31, 2013