Linear Algebra (Dover Books on Mathematics) by Georgi E. Shilov (PDF)

Description

In this volume in his exceptional series of translations of Russian mathematical texts, Richard Silverman has taken Shilov’s course in linear algebra and has made it even more accessible and more useful for English language readers.Georgi E. Shilov, Professor of Mathematics at the Moscow State University, covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional algebras and their representations, with an appendix on categories of finite-dimensional spaces.The author begins with elementary material and goes easily into the advanced areas, covering all the standard topics of an advanced undergraduate or beginning graduate course. The material is presented in a consistently clear style. Problems are included, with a full section of hints and answers in the back.Keeping in mind the unity of algebra, geometry and analysis in his approach, and writing practically for the student who needs to learn techniques, Professor Shilov has produced one of the best expositions on the subject. Because it contains an abundance of problems and examples, the book will be useful for self-study as well as for the classroom.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book arrived quickly and was in great condition.

⭐You will find a list of errata posted on reddit.

⭐Shilov leaves the reader with deep intuition for the subject, forgoing the random assortment of special cases and applications so common in contemporary texts. The treatment toward the middle of the book of linear forms and change of basis paves the way for bilinear and quadratic forms (culminating in categorization of quadratic surfaces by their topology)in the second half of the book, with a final chapter on representation of abstract algebras and an appendix on categories of spaces. Masterpiece.

⭐I left a review of this book several years ago and am completely rewriting it. I previously gave it 5 stars but I can’t really justify that anymore and have to lower it to 4. I still really like it and would absolutely recommend it for a second course on linear algebra; as a first course it would be a complete disaster (something along the lines of Rudin’s Principles of Mathematical Analysis as a first exposure to calculus for someone who’s never heard of a limit before). There’s a lot of material built up step-by-step and I found almost all of the proofs to be mostly very clear, provided of course that you’ve already been exposed to the basics and can deal with the terseness of the book (though Shilov relies pretty heavily on rather ugly equations with a lot of explicit sub/superscripts where matrix equations would be cleaner). There are the usual typos that occur in any math book (a decent amount of them, actually), but I haven’t found any major ones that hinder understanding. The first chapter on determinants is amazing and covers just about all you need to know about them, although the definition of the determinant may be unfamiliar and no motivation is given for it at first. Later chapters shed some light on this, and anyway the reader will quickly realize that this strange definition has all the properties they’re familiar with. Chapters 2-4 also cover mostly familiar material but possibly from a slightly different perspective, and no doubt some of the material will be new.The early sections of Chapter 5 are the first example of why I lowered the rating. He defines a transformation matrix between bases and then uses that to get a transformation matrix between coordinates, but he never actually says what that matrix does. It is NOT the matrix that multiplies the column vector consisting of the old coordinates to give you the column vector of the new coordinates. Now, he never actually claims that it is, so it’s not really an error per se, but then what is this transformation matrix he defines on p.122? It’s the transpose of the matrix that actually acts on the coordinates in the way described above, but why would we care about the transpose? It makes no sense to me.Another baffling choice he makes is in chapter 7 (top of p.197 to be precise) where he suddenly switches the convention of having the columns of the matrix of a linear transformation represent the images of the basis vectors to having the rows represent the images. WHY? Attention is drawn to this switch but no reason for it is given. This means that if you want to stick with the convention used earlier (and why wouldn’t you? EVERY linear algebra book I’ve seen uses it), you have to swap a bunch of indices and swap certain instances of the words “row” and “column” throughout the chapter. Also in chapter 7 he uses the letter A to represent entirely different functions within the same problem. A could be a fixed bilinear form and an arbitrary linear transformation within the same sentence. Or A could be a bilinear form acting in K’ and another bilinear form acting in another (although isomorphic) space K”. I know what he means and I can follow the proofs, which are ultimately the things that really matter, but there’s no good reason for such sloppiness. A final criticism of chapter 7 is that not once in the entire chapter does he explicitly state that a bilinear form B(x,y) can be written as x^TBy, even though using this matrix form would make many of the proofs much cleaner.The rest of the book, of which I’ve gone through most but not all, is similar. The general progression of topics is interesting and logical, the proofs are understandable, and it’s enjoyable to work through, but be prepared to write a lot of marginal notes and/or cross out and replace some words here and there.

⭐Got me over the hump when I could not remember how to do deal with matrix operations. I did not go non linear over this book.

⭐For a little over 10 bucks, it’s worth getting this VERY DENSE, terse Russian – translated version of LA. If you seriously need to study Linear Algebra, you’d also need an excellent and CLEAR book– a couple best practices examples would be 1.

⭐($12) and 2.

⭐.Shilov covers a wide variety of topics, both basic and advanced, but the language is abbreviated, and the notation cumbersome, with almost 400 pages of mice type equations. He’ll make a statement like, “of course you’ll need to convert this to determinant form” and instead of explaining it, give another equally dense set of equations. When you use equations and proofs to “teach” other equations, it assumes a LOT of previous knowledge– NOT for beginners!If you already have a good grounding in LA, this is a gem well worth the low price, as it covers many unique proofs and “older” roots (1971). This doesn’t make it outdated, except that of course you won’t see topics like quantum computing covered, but will get many angles not seen in more up to date texts. Add it to your library for advanced work, but don’t get it to learn LA from square one– it packs too much into each page with too little explanatory diagrams, applications, examples or descriptions. It basically assumes you’re very well grounded in pure math, formula notation, and proofs.BTW, if you’re a professor or other pro in LA, the index alone is worth the price– works very well as a reference. Wish there was something similar for ODE’s!

⭐I’m a first year maths student at nottingham university and bought this to aid one of my modules. I haven’t worked through most of it but from flicking through it looks like about half of it is on stuff i have covered this year. Working through it it is helpful because it often approaches things in different ways to how we have done in lectures. I only bought it about a month before exams start and wish i had bought it earlier as i dont think i have time to take advantage of it properly. It also has problems for the reader to tackle at the end of every chapter.

⭐Excellent book!

⭐Lo recomiendo como un libro para aprender a fondo el álgebra lineal, recomendable para matemáticos, físicos, y para aquellos que quieran aprender a fondo la materia, a pesar de tener ejercicios, estos no están enfocados en hacer calculitos, no es libro de ingeniería es un libro teórico, pero no es complicado, por el contrario. Dicho lo anterior mientras que libros como grossman solo se enfocan en la parte mecánica de resolver ejercicios numéricos Shilov se enfoca en demostrar y mostrar las propiedades y conceptos del álgebra lineal. Su redacción es una de las más claras y fáciles de leer(como matemático) ya que su formalidad no deja lugar a dudas, es consiso y va al punto sin hacer mucho verbo (verbose). Se nota que es un libro de la Unión soviética. Personalmente lo he usado para estudiar temas no solo de álgebra lineal si no de vectorial, ecuaciones diferenciales y un poco de análisis matemático, es decir tiene una gran variedad de temas, es sorprendente la cantidad de cosas que explica, todo perfectamente numerado y oye lo cual fácilmente citable, por lo que además sirve mucho para referencias y encontrar temas para documentar. Personalmente estudie al inicio con el friedberg, Hoffman , Sheldon Axler, sin duda mi favorito es este. El único punto es que se necesitará estudiar que es un determinante antes de empezar este libro, su relación con el área de un paralelogramo(es decir el determinante mostrará el cambio de “volumen” de las bases a partir de cada vector), y lo que es un sistema de ecuaciones, después de demostrar esto sigue este libro, ya que no incluye dicha introducción(que no son muchos temas). En resumen este libro es demasiado intuitivo formal fácil de leer, y con muchos temas incluye ejercicios en su mayoría conceptuales, se ha convertido en uno de mis 20 libros favoritos entre mi enorme colección.# If you dream to be an elite computer programmer…It is really well written classic univ level algebra.One important fact is that you need to be familiarwith math specialists and multivariable to understand it.there is more emphasis on one math in the computer science field.Maybe the definition of computer programmer is being changed.software engineer these days should be good at not only discrete mathbut also other disciplines of math.

⭐This is a great book to learn linear algebra from. It builds the field from the ground up, so you can really understand what’s happening under-the-hood. It’s very theoretical, though. You should probably combine it with something like Linear Algebra and Its Applications (by G. Strang) – for a more practical perspective.

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