### 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.