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 field 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 finitely 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 finite.