The steinitz exchange theorem and its applications.

Abstract

It seems that during the last decades, no research was done which is related to the Steinitz exchange theorem. However, the generalised Steinitz exchange theorem has been investigated in books and articles . The generalized Steinitz exchange theorem is not a theorem of linear algebra but for reaching generalization of the Steinitz exchange theorem which has applications for example in eld theory, in the theory of abelian groups and in module theory. The objective of this study was to prove the Steinitz exchange theorem of linear algebra for arbitrary vector spaces over arbitrary division rings. Nearly all books on linear algebra which have the Steinitz exchange theorem explicitly state and prove this theorem only for nitely generated vector spaces. Only one exception can be found. In another source, the Steinitz exchange theorem is proved under the additional assumption, that the linearly independent subset is nite. In this study the exchange theorem of Steinitz is proved in full generality with the means of linear algebra. The statement of the theorem of Steinitz is a statement of the following type: under certain conditions there exists a set with certain properties.The question when this set is uniquely determined could be completely solved. In addition, an application of the theorem of Steinitz is presented. This is the classical application which was given already by Gra mann: Any two bases of a vector space are equipotent. The rst chapter is about the basic concepts of the study. The second chapter reviews the relevant literature and outlines the methodology used in the study. The literature review is mainly about the generalized theorem of Steinitz, but also include the versions of the Steinitz exchange theorem found in books of linear algebra. The third chapter presents the results of the study with proofs. The study is concluded in the last chapter with proposals for further study.