Near vector spaces

De Bruyn, Aletta (1990)

Thesis (M. Sc.) -- University of Stellenbosch, 1990.

Thesis

ENGLISH ABSTRACT: The preliminary material in Chapter 1 is included in order that this work be reasonably self-contained. The main aim of this thesis is to give an exposition of the theory of near vector spaces, as introduced by J. Andre [1]. This is done in the second chapter. Concepts such as an F-group and its quasi-kernel are defined and investigated. A near vector space is then defined in terms of these notions, after which the theory of near vector spaces is developed. Three examples are given in Chapter 3. Their quasi-kernels are investigated and, in each case, it is shown that the near vector space under consideration is not a vector space. Finally, in the fourth chapter, certain known results concerning linear transformations of finite-dimensional vector spaces are recalled. These notions are then investigated for certain near linear transformations of a 2-dimensional near vector space.

AFRIKAANSE OPSOMMING: Die inleidende eerste hoofstuk, in die besonder die afdeling oor afhanklikheidsrelasies, is hoofsaaklik vir verwysingsdoeleindes. Die hoof doel van die tesis is om 'n uiteensetting van die teorie van byna-vektorruimtes, soos cleur .J. Andre [1] ontwikkel, te gee. Dit word in Hoofstuk 2 gedoen. Begrippe soos 'n F-groep en sy kwasi-kern word ondersoek. Byna-vektorruimtes word dan in terme hiervan gedefinieer. Hierna word die teorie van byna-vektorruimtes aangebied. In Hoofstuk 3 word drie voorbeelde van byna-vektorruimtes gegee. Hulle kwasi-kerns word ondersoek en daar word, in elke geval, aangetoon dat die betrokke byna-vektorruimte nie 'n vektorruimte is nie. Sekere bekende konsepte rakende lineêre transformasies van eindige vektorruimtes word in die vierde hoofstuk bespreek. Hierdie begrippe word dan ondersoek in die geval van sekere byna-lineêre transformasies van 'n 2-dimensionale byna-vektorruimte.

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