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Linear Algebra (Dover Books on Mathematics), by Georgi E. Shilov
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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 space. Problems with hints and answers.
- Sales Rank: #61209 in Books
- Published on: 1977-06-01
- Released on: 1977-06-01
- Original language: Russian
- Number of items: 1
- Dimensions: 8.38" h x .78" w x 5.64" l, .91 pounds
- Binding: Paperback
- 400 pages
Language Notes
Text: English, Russian (translation)
Most helpful customer reviews
140 of 145 people found the following review helpful.
Excellent Linear Algebra Text
By S. Murphy
This is a solid book, but requires a degree of mathematical maturity. Like many of the Dover publications of translated Russian mathematical texts, the book is clearly written, with good proofs that are easy to follow, lots of useful examples, and solutions to problems are given at the end of the book.
Readers should note that the author is a noted Russian mathematician, a former professor of mathematics at Moscow University, one of great centres of mathematical research and teaching in the world. Shilov collaborated with many important mathematicians such as Kolmogorov and Gelfand. If you have read any of Kolmogorov or Gelfand's excellent Dover books, then the style of this book is very similar to those.
77 of 78 people found the following review helpful.
Outstanding Book
By Alexander C. Zorach
I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them. However, there is no book I have found that has such a deep and clear exposition of determinants. The first chapter alone makes this book worth buying.
However, there's an incredible amount of material in this book, and the later chapters are just as valuable. This is a dense book, but it is fairly easy to read once you get used to the style. I would recommend it to anyone learning linear algebra for the first time, as well as to people who want a deeper understanding or a different perspective.
Like I said before, this book is particularly useful when combined with a complementary text such as Axler, which provides a completely different approach to the subject. This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra. And plus: you can't complain about the price!
42 of 44 people found the following review helpful.
An Excellent Second Book on the Subject
By Jason Dowd
If you are trying to learn linear algebra for the first time, avoid this book. You will be better served by Schaum's Outline of Linear Algebra Fourth Edition (Schaum's Outline Series). However, if you already have a working knowledge of the basics, have realized the central importance of this subject, and are trying to reach a deeper understanding of it, this is the book for you.
Here is a brief rundown of the contents:
Chapter 1 is on the theory of determinants. It is excellent and forms the basis of the entire book. It also immediately shows the sole focus of this book: finite dimensional linear algebra.
Chapter 2 is on linear spaces. Here as in his other books, Shilov moves to the abstract very early in his treatment. Anyone reading this book should be prepared by at least having the background of Schaum's Outline of Set Theory and Related Topics. It is noteworthy that this book does not even admit the existence of infinite dimensional vector spaces. Indeed, a basis for a vector space must be finite by definition. This is quite nonstandard, but again shows the focus: finite dimensional linear algebra.
Chapter 3 treats systems of linear equations. This chapter moves us back to the concrete, but again this book is only for readers with a firm grasp of the basics. Gaussian elimination is never even mentioned, but Cramer's rule is discussed in detail and used frequently for the rest of the book as a powerful theoretical tool. This might lead the uninitiated to believe that Cramer's rule is actually how systems of linear equations are solved when in fact this is almost never actually the case. Cramer's rule quickly becomes computationally infeasible while Gaussian elimination scales much more gently. And this again shows the focus of this book: theoretical understanding rather than practical manipulation. Sadly, even the lack of scaling for Cramer's rule is not mentioned.
Chapter 4 treats linear functions. This chapter starts with the abstract notion of linear operators and then moves to matrix operations. Eigenvalues and eigenvectors are also covered in this chapter.
Chapter 5 treats change of basis and transformation properties. It has an optional section on tensors which is quite good, but notationally inconvenient.
Chapter 6 is on canonical forms. In many ways this chapter is the theoretical centerpiece of the book. The work requires a good elementary understanding of polynomial algebra, and proceeds at quite an abstract level. But it is excellent for anyone really wanting a firm grasp of this central topic.
Chapter 7 is on bilinear and quadratic forms. Again, it is quite abstract but excellent.
Chapter 8 is on Euclidean spaces and contains one of the highlights of the book in its treatment of the method of least squares. This one section is enough to make the entire book worthwhile.
Chapter 9 is on Unitary spaces. Key stuff for anyone wanting to learn quantum mechanics.
Chapter 10 is on quadratic forms in Euclidean space and contains key material for the student of classical mechanics.
Chapter 11 is on finite dimensional algebras and their representations. This entire chapter is marked as optional and is considerably harder going than most of the material up to this point. But again, it is just excellent and pretty good preparation for anyone interested in group theory and group representations.
Finally, Appendix A is on categories of finite dimensional spaces. This is definitely the most abstract part of the book, but is very well done and again, optional.
There are quite a few typos in this book and some exasperating language errors in some of the proofs like "not more than" instead of "at least as many", and the reader will need to keep a sharp lookout. Also, I wasn't thrilled about some of the author's notational choices.
Problems follow at the end of every chapter. They are generous in number and range from easy to challenging. Even better, many of the problems have solutions or at least strong hints in the back of the book.
This is an excellent book for anyone interested in mastering the theory of finite dimensional linear algebra, and that should include just about any student of mathematics, the physical sciences, the social sciences, engineering, finance, or computer science.
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