  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.

General Mathematics