Linear Algebra Over Division Ring System of Linear Equations
“Linear Algebra Over Division Ring System of Linear Equations” by Aleks Kleyn, published by CreateSpace Independent Publishing Platform on June 16, 2012, is a focused exploration of linear algebra in the context of division rings. This 60-page book presents the properties of systems of linear equations over division rings, highlighting the similarities and differences compared to systems over fields. The author delves into the implications of noncommutativity in products and the role of matrices, offering insights into the structure of birings and their associated vector spaces.
Readers will find a detailed examination of vector spaces alongside systems of linear equations, emphasizing their interrelated properties. The book discusses the group of automorphisms of a vector space and introduces concepts of passive and active representations. By studying vector spaces over division rings, the text reveals new aspects of the relationship between transformations, providing a clearer understanding of these mathematical constructs. This edition is presented in English and is designed for those interested in pure mathematics and its applications in science and math.
Official synopsis Publisher
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.
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