Titlebook: Linear Algebra and Geometry; Igor R. Shafarevich,Alexey O. Remizov Textbook 2013 Springer-Verlag Berlin Heidelberg 2013 groups, rings, mod

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od introduction to the subject.Numerous examples and applica.This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory

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Matrices and Determinants,symmetric multilinear function of the rows. Using some basic elements of permutation theory, we continue to study the properties of determinants; in particular, we derive explicit formula for determinants. Finally, we define the rank of a matrix and the main operations on matrices (sum, product, inverse matrix) and investigate their properties.

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Linear Transformations of a Vector Space to Itself,lizable. Analogously, for a real vector space, we obtain necessary and sufficient conditions for a linear transformation to be block-diagonalizable. Finally, the notion of orientation of a real vector space is considered.

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Quadratic and Bilinear Forms, are established. Complex, real, and Hermitian forms are investigated in greater detail. For illustration of the obtained results, we consider an application of Sylvester’s criterion to algebraic equations (necessary and sufficient conditions for a real polynomial to have only real roots).

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Hyperbolic Geometry,rief historical overview is given. At the end of the chapter, some geometric notions (distance, angle, etc.) are introduced, and basic facts of hyperbolic geometry are established. A brief discussion of elliptic geometry is also presented.

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