Linear Algebra Georgi Shilov

[Venice]

Igraduated from high school three years ago and I am plannig on going to university next year to study mathematics.Since I have a year to prepare, I would like to learn some first year university subjects, like linear algebra. Is Shilov's "Linear Algebra", suitable for this purpose or is it too advanced ?

I haven't bought this book, but I have access to a copy of it. On a cursory read, I think it isn't too advanced and its choices of topics are about right. It covers slightly more topics than one shall see in a typical modern introductory course that focuses only on the theory of linear algebra.

Whether the book is suitable for beginners depends very much on the readers. There are books (like Rudin's Principles of Mathematical Analysis or Spivak's Calculus on Manifolds) that are praised by many people but also found too hard by equally many. To me, Shilov's exposition looks clear, efficient and rigourous. It is harder (but not to a large extent) than most modern introductory texts on linear algebra, but much easier to read than Hoffman and Kunze's classic, Linear Algebra.

Shilov introduces and discusses determinants in chapter 1. This is usually one of the most difficult topics in an introductory course. If you can get past chapter 1 without difficulty, you should be able to understand the whole book (sans the chapters marked by asterisks).

However, given that the book was published in the 1970s, those readers who are accustomed to the new breed of more "conversational" texts may dislike its style. They may also find it lacking motivations or practical applications.

If you are currently interested only in the theory (but not applications) of linear algebra, I think Berberian's Linear Algebra is another inexpensive and viable choice for private study. Jim Hefferon's Linear Algebra also worths a look. It does cover some applications and it is a favourite text of Darij Grinberg, a well respected user of this site. The electronic copy is free and the paper copy is cheap.

Strang's Linear Algebra is a mathematical discipline that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical concepts to solve problems related to linear systems and transformations.

Strang's Linear Algebra is unique in its approach to solving linear problems. It focuses on understanding the underlying concepts and geometrical interpretations of linear equations, rather than just manipulating symbols and numbers. This makes it more intuitive and applicable to real-world scenarios.

Strang's Linear Algebra has a wide range of applications in fields such as engineering, physics, economics, and computer science. It is used to solve problems related to optimization, data analysis, image processing, and machine learning, among others.

Like any other mathematical discipline, Strang's Linear Algebra can be challenging to learn, especially for those without a strong mathematical background. However, with dedication and practice, it can be grasped by anyone, and its practical applications make it worth the effort.

Yes, you can learn Strang's Linear Algebra on your own through various resources such as textbooks, online courses, and video lectures. It is recommended to have a basic understanding of algebra and calculus before delving into Strang's Linear Algebra.

This text can be viewed as an introduction to reading mathematics texts; The initial proofs are on arithmetic and properties of linear equations, and are more approachable than the manipulations of functions typical of analysis texts. It is difficult to overstate the universal importance of this subject, but it can be seen through reading. Linear algebra is the basis of all quantities of interest in physics, geometry, number theory, and the same techniques appear in engineering disciplines through physics, numerical computing, or machine learning.

Another similarly famous Russian text on linear algebra would be Kostrikin and Manin's Linear Algebra and Geometry but better as a second casual read since it is more advanced. It contains more information about the interaction between linear algebra and quantum mechanics and relativity, and classcial projective geometry such as the Hopf fibration and Plcker coordinates. Projective geometries are treated concretely here, but are eventually characterized much more abstractly and generally in algebraic topology.

The author, Georgi E. Shilov, has published several books on topics including linear algebra and mathematical analysis. This is translated from Russian by Richard A. Silverman. The material is well written in a clear unambiguous style. The book is suitable as a standalone text. The logic builds nicely throughout the book.

Overall the book provides a solid treatment of the theory of linear spaces and focuses heavily on theory over practical computation and algorithms. In this way it is a little different to more contemporary texts which have a stronger emphasis on computation and algorithms.

This book starts with a thorough treatment of determinants. In this way it is very different to other books that tend to cover this topic later, and in much less depth. This first chapter is the reason to get this book.

The text then continues on a logical path covering the theory of linear spaces, then linear operators. The sixth chapter on the canonical form of a linear operator has a strong focus on the Jordan canonical form. This might again set this text apart from others that do not focus on the Jordan form.

The document discusses several topics related to linear algebra:1) It asks to determine if several matrices are in reduced row echelon form, row echelon form, or neither. 2) It asks to solve systems of linear equations using Gaussian elimination and Gauss-Jordan elimination.3) It asks to find the inverse of a matrix using elementary row operations on an augmented matrix. 4) It discusses row equivalence and finding a sequence of row operations to transform one matrix into another.5) It asks to find conditions for systems of linear equations to be consistent.Read less

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