Sergei Kurgalin - Algebra and Geometry with Python

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The rapid development of computing places special demands on the training of young specialists in this field. The complexity of the problems that must be solved to satisfy the needs of science, technology, industry and the economy also grows, simultaneously with the rapid growth in the power of modern computer systems. In this connection, we believe that it is essential that computing specialists have fundamental knowledge about the development of the related mathematical frameworks and about how to create methods for solving the above problems.

Algebra and geometry are deemed important areas whose ideas and results are actively used in the development of information systems, as well as in software developed for business projects. The basic notions of algebra are numerical matrices and the methods for working with matrix algorithms. They may be used extensively in scientific and technical problems and in the game industry. The rapid development of game technologies, as well as augmented and alternative reality technologies, means that we must pay special attention to university courses in analytical geometry and linear algebra, pattern properties in 3D space and fast algorithms for working with two- and three-dimensional objects.

Another promising area of application for linear algebra algorithms that has seen rapid development in recent years is Big Data. Analysis of extremely large arrays requires not only knowledge and use of the known methods, but it also issues the challenge to develop new approaches and high-performance algorithms.

This textbook is an introduction to linear algebra and analytical geometry for higher-education students in the natural sciences. It is based on the courses Algebra and Geometry, Analytical Geometry and Fundamental and Computer Algebra, which are taught to first-year students of the Faculty of Computer Sciences at the Voronezh State University. The teaching is meant for theoretical training, as a supplement to the existing textbooks, for practical and laboratory classes, and also for self-study. Going forward, the terms Algebra and Linear Algebra will be considered equivalent, as well as Geometry and Analytical Geometry.

The authors have attempted to lay the material down in the most comprehensible form while not sacrificing strictness in definitions and theorems. The statements (theorems, properties) are accompanied with proofs, or references to specialist literature for advanced study of the materials.

The fundamentals of algebra and geometry are presented in the form most suitable for future specialists in computing. We have considered the basic algorithms for working with matrices, vectors and systems of linear equations. The theoretical material contains solutions of most types of problems and is supplemented with plenty of analysed examples. The end of an example is designated by the symbol . Each chapter ends with problems for self-study. Many of them are provided not only with full answers but also with detailed solutions. The asterisk sign (*) marks the advanced (enhanced complexity) problems.

Apart from the sections traditionally included in algebra and geometry courses, one of the chapters is devoted to the mathematical fundamentals of the modern section of cryptography, namely elliptic curve cryptography. The availability of this chapter will be a connecting link between the mathematical courses and methods applied in practice by the application software developers.

The section about quantum computing is devoted to one of the examples of the application of algebra. It demonstrates that the notions of linear algebra are used for constructing new algorithms, whose computation capacity exceeds the existing ones considerably.

Let us briefly summarize the content of this textbook. The first four chapters are devoted to classical divisions of linear algebra; they consider matrices and determinants, and systems of linear equations; definitions are given for the notion of vector space and the fundamental solution of a homogeneous system. The next few chapters introduce the fundamentals of vector algebra and the coordinate method on a plane and in a 3D space. The following subjects are considered: vectors in three-dimensional space, the equation of a line on a plane, the equation of a plane in space and the equation of a line in space. Second-order curves are analysed. Material on elliptic curves is usually not included in a traditional algebra and geometry course. However, its presence in this book, in our opinion, contributes to a deeper understanding of the methods of linear algebra and analytical geometry and provides an example of the implementation of such methods for solving problems in theoretical and practical cryptography.

We use the Python programming language for illustration of the considered algorithms. This allows us to familiarize readers with implementation at the initial stage of study. Python was selected because it is a universal and widely used general-purpose programming language, suitable for the successful realization of numerical algorithms; Python is a continuously evolving language; and many of its realizations are open source. Python has the necessary tools to automatically check for the errors that might appear in the program code in the process of its creation. The availability of a great number of additional libraries (such as