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Lectures on Linear Algebra over Skew Field

book
Language:
Author: Aleks Kleyn
Url: http://babbage.sissa.it/abs/math/0701238
Format: Ps, Pdf
Year: 2007
Category: Linear Algebra
Pages: 70
Clicks: 2855

Description
In this book i treat linear algebra over skew field. A system of linear equations over a skew field 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 skew field uncovers new details in the relationship between passive and active transformations, makes this picture clearer.(english and russian version).

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