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An Introduction to Linear Algebra and Tensors, Revised Edition, by M. A. Akivis, V. V. Goldberg
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The present book, a valuable addition to the English-language literature on linear algebra and tensors, constitutes a lucid, eminently readable and completely elementary introduction to this field of mathematics. A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention. The treatment is virtually self-contained. In fact, the mathematical background assumed on the part of the reader hardly exceeds a smattering of calculus and a casual acquaintance with determinants.
The authors begin with linear spaces, starting with basic concepts and ending with topics in analytic geometry. They then treat multilinear forms and tensors (linear and bilinear forms, general definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again basic concepts, the matrix and multiplication of linear transformations, inverse transformations and matrices, groups and subgroups, etc.). The last chapter deals with further topics in the field: eigenvectors and eigenvalues, matrix ploynomials and the Hamilton-Cayley theorem, reduction of a quadratic form to canonical form, representation of a nonsingular transformation, and more. Each individual section — there are 25 in all — contains a problem set, making a total of over 250 problems, all carefully selected and matched. Hints and answers to most of the problems can be found at the end of the book.
Dr. Silverman has revised the text and numerous pedagogical and mathematical improvements, and restyled the language so that it is even more readable. With its clear exposition, many relevant and interesting problems, ample illustrations, index and bibliography, this book will be useful in the classroom or for self-study as an excellent introduction to the important subjects of linear algebra and tensors.
- Sales Rank: #1270423 in Books
- Published on: 2010-10-18
- Released on: 2010-09-20
- Original language: Russian
- Number of items: 1
- Dimensions: 8.24" h x .39" w x 5.62" l, .48 pounds
- Binding: Paperback
- 192 pages
From the Back Cover
The present book, a valuable addition to the English-language literature on linear algebra and tensors, constitutes a lucid, eminently readable and completely elementary introduction to this field of mathematics. A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention. The treatment is virtually self-contained. In fact, the mathematical background assumed on the part of the reader hardly exceeds a smattering of calculus and a casual acquaintance with determinants.
The authors begin with linear spaces, starting with basic concepts and ending with topics in analytic geometry. They then treat multilinear forms and tensors (linear and bilinear forms, general definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again basic concepts, the matrix and multiplication of linear transformations, inverse transformations and matrices, groups and subgroups, etc.). The last chapter deals with further topics in the field: eigenvectors and eigenvalues, matrix polynomials and the Hamilton-Cayley theorem, reduction of a quadratic form to canonical form, representation of a nonsingular transformation, and more. Each individual section — there are 25 in all — contains a problem set, making a total of over 250 problems, all carefully selected and matched. Hints and answers to most of the problems can be found at the end of the book.
Dr. Silverman has revised the text and numerous pedagogical and mathematical improvements, and restyled the language so that it is even more readable. With its clear exposition, many relevant and interesting problems, ample illustrations, index and bibliography, this book will be useful in the classroom or for self-study as an excellent introduction to the important subjects of linear algebra and tensors.
Unabridged and unaltered republication of revised English edition originally titled Introductory Linear Algebra, 1972.
About the Author
Maks A. Akivis is Professor of Mathematics at the Ben-Gurion University of the Negev in Beer-Sheva, Israel, and at the Moscow Institute of Steel and Alloys in Russia.
Vladislav V. Goldberg is Distinguished Professor of Mathematics at the New Jersey Institute of Technology in Newark.
Dr. Akivis and Dr. Goldberg are the authors of numerous papers, many of which they wrote jointly. They are the authors of the book Tensor Calculus and the monograph Projective Differential Geometry of Submanifolds. In addition, Dr. Akivis is a coauthor of the monograph Geometry and Algebra of Multidimensional Three-Webs and the book Elie Cartan (1869-1951), and Dr. Goldberg is the author of the monograph Theory of Multicodimensional (n+1)-Webs.
Most helpful customer reviews
23 of 24 people found the following review helpful.
A decent book with lots of exercises
By A MATH NERD
This book is not the best linear algebra book I've come across, but there are a lot of good things about it. The proofs are all very clear, and there are lots and lots and lots of good exercises. Something I see with a lot of math books on the same topic is that they often have a lot of exercises in common-not usually exactly the same, but difering only by a few numbers or words. But many of the exercises in this book, particularly in the early chapters on dimension, cross product, and dot product, I have not seen in any other book. The one thing about this book is that there really is not a huge amount of non-exercise text-though what there is is well-written. So maybe this would work best as a supplement to another book. One thing that can be said about that book is that, in the division of linear algebra books into computational or abstract algebra books, this book is somewhere in the middle. It starts with the axioms of a vector space, but most of the text concerns only 3-dimensional euclidean geometry-though many(but not all!) of the proofs carry over to higher dimensions without change. Also, the inclusion of so much material on the cross product-which is really useful only in applications to physics(as far as I know), not in abstract mathematics, is another unique feature of this book. Now, this book does not contain things like Gaussian elimination, but it is still not all that abstract, compared to many other books, at least. Also, this book is very short. It covers all the basics, but simply ignores some topics such as tensor products(necessary to a good treatment of tensor products, not messy and index-laden like the one here), exterior products, Jordan normal form, as well as much about what happens if the base field isn't R-in particular, anything about Hermitian or unitary matrices(Unless my memory has failed me-I don't have the book at hand to be sure these things were never mentioned, but am pretty sure).
0 of 1 people found the following review helpful.
Five Stars
By MOISES ARANDA-SILVA
it is a concise book.
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