Lectures on Linear Algebra over Division Ring

TL;DR

This book explores linear algebra over division rings, highlighting how noncommutativity introduces new structures and relationships, extending classical linear algebra concepts to a broader algebraic context.

Contribution

It develops the theory of vector spaces, linear maps, and tensor products over division rings, including new insights into automorphisms and representations.

Findings

Solutions form right or left vector spaces depending on the system type

Introduces the concept of D vector spaces and their linear maps

Defines tensor products of rings and D vector spaces

Abstract

In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a new picture. Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. As in the commutative case, solutions of a system of linear equations build up right or left vector space depending on type of system. We study vector spaces together with the system of linear equations because their properties have a close relationship. As in a commutative case, the group of automorphisms of a vector space has a single transitive representation on a frame manifold. This gives us an opportunity to introduce passive and active representations. Studying a vector space over a division ring uncovers new details in the relationship between passive and active transformations, makes this picture clearer. Considering of twin representations of division ring in Abelian group leads to the concept of D vector space and their linear map. Based on polyadditive map I considered definition of tensor product of rings and tensor product of D vector spaces.