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Alexei I. Kostrikin and Yu. Jensen and Helmut Lenzing This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details. All rights reserved. Linear algebra and geometry. Algebra, Linear 2. Geometry 1. Title II. Manin, IU. Iurii Ivanovich , In this case our task is facilitated by the fact that we are discussing the paperback edition of our textbook.

One can look at the subject matter of this course in many different ways. For a graduate student, linear algebra is the material taught to freshmen. For the professional algebraist, trained in the spirit of Bourbaki, linear algebra is the theory of algebraic structures of a particular form, namely, linear spaces and linear mappings, or, in the more modern style.

From a more general viewpoint, linear algebra is the careful study of the mathematical language for expressing one of the most widespread ideas in the natural sciences - the idea of linearity. The most important example of this idea could quite possibly be the principle of linearity of small increments: almost any natural process is linear in small amounts almost everywhere.

This principle lies at the foundation of all mathematical analysis and its applications. The vector algebra of three-dimensional physical space, which has historically become the cornerstone of linear algebra, can actually be traced back to the same source: after Einstein, we now know that the physical three-dimensional space is also approximately linear only in the small neighbourhood of an observer.

Fortunately, this small neighbourhood is quite large. Twentieth-century physics has markedly and unexpectedly expanded the sphere of application of linear algebra, adding to the principle of linearity of small increments the principle of superposition of state vectors. Roughly speaking, the state space of any quantum system is a linear space over the field of complex numbers. MANIN The selection of the material for this course was determined by our desire not only to present the foundations of the body of knowledge which was essentially completed by the beginning of the twentieth century, but also to give an idea of its applications, which are usually relegated to other disciplines.

Traditional teaching dissects the live body of mathematics into isolated organs, whose vitality must be maintained artificially. During the last half-century the language and fundamental concepts were reformulated in set-theoretic language; the unity of mathematics came to be viewed largely in terms of the unity of its logical principles. We wanted to reflect in this book, without ignoring the remarkable achievements of this period, the emerging trend towards the synthesis of mathematics as a tool for understanding the outside world.

Unfortunately, we had to ignore the theory of the computational aspects of linear algebra, which has now developed into an independent science. Based on 1.

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