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An Introduction to Linear Algebra and Tensors, Revised Edition download epub

by M. A. Akivis,V. V. Goldberg,Richard A. Silverman


Epub Book: 1660 kb. | Fb2 Book: 1920 kb.

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.

Hints and answers to most of the problems can be found at the end of the book.

A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention. Hints and answers to most of the problems can be found at the end of the book.

Additional topics include multilinear forms, tensors, linear transformation, eigenvectors and eigenvalues, matrix polynomials, and more. More than 250 carefully chosen problems appear throughout the book, most with hints and answers. To read this book, upload an EPUB or FB2 file to Bookmate.

Akivis, M. A. (Maks Aĭzikovich). Physical Description. iii-vii, 167 p. : ill.

A. Akivis, V. V. Goldberg, Richard A. Silverman. A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention.

A special merit of the book, reflecting its lineage, is its free use of tensor notation, in particular the Einstein summation convention. Each of the 25 sections is equipped with a problem set, leading to a total of over 250 problems. Hints and answers to most of these problems can be found at the end of the book. (Maks Aĭzikovich); Golʹdberg, V. (Vladislav Viktorovich); Silverman . (Vladislav Viktorovich); Silverman, Richard A. Publication date. in 1972 under the title: Introductory linear algebra. Includes bibliographical references (page 161) and index. 1. Linear spaces - 2. Multilinear forms and tensors - 3. Linear transformations - 4. Further topics.

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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.

Comments: (4)

Halloween
A very good book for students and professionals alike, who want to strengthen their knowledge of linear algebra and tensors.
MrDog
This approach to tensors, a translation from the Russian, offers a few very clear advantages over its American counterpart, the most important being that it is a rather painless introduction.

By seamlessly integrating analytic geometry, linear algebra, and vector calculus, no abrupt gaps in understanding appears. The authors introduce “linear” and “bilinear” forms early and without the benefit of topological concepts, which one could argue results in a wash, when compared to introducing them later — as is done in US academies.

In any case, and here is the crowning benefit in my view: here (but not in US texts) the end point seems clear from the start: to use vector calculus in a functional way to establish the mathematical machinery needed to deal with differential curves and surfaces — that is to say, ultimately, to get into Differential Geometry as quickly as possible, and with all four feet.

The rest is just mathematical details — clean-up operations as it were, like proper indexing, properly following the rules of new geometrical and algebraic entities up through the more advanced topics and concepts. Then making sure that as you ascend the ladder of complexity to topics such as algebraic operations on Tensors, that all the new rules are recognized and obeyed precisely.

On the way up, the notation can get hairy, but here the Occum’s Razor approach seems to work best: the less complex the notation, the better.

As one would expect, as we move up the ladder to more intricate mathematical concepts, topics and entities, to more functional complexity, of course the rules governing them become more complex too — as well as more subtle and more nuanced. However, since these authors carefully selected all of the examples in a graduated fashion, giving us clear warning when notation and indexing are ratcheted-up, each new step up in complexity is manageable and thus is not in the least bit frightening. I like this approach because it is more difficult to get lost in the indexing.

Thus, there is nothing scary about this book. However, I suspect modern Western Mathematicians may think this book does not go quite far enough.

On that issue I am unqualified to make an informed assessment. Except to note that overlapping topics between Russian and Western mathematical approaches, even when headed towards the same end point, reorder the topics presumably to suit their respective styles.

For instance, Eigenvectors and Eigenvalues are introduced quite early in American discourse and quite late here. Conversely, (and as noted above) linear and bilinear forms are introduced early here and late in most American texts.

A final note is that the Russian approach (cira 1972, the date of publication of this book), more closely resembles the contemporary engineering approach taken in the U.S.: Proofs are eschewed and functional operators are insinuated into the formulas early on and without fuss or fanfare. Pure Mathematicians are sure not to like that, but arguably, it facilitates progress.

Finally, this is obviously an “old style” introduction to Tensors, but it works for me. Five Stars
Hap
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).
Walianirv
it is a concise book.
An Introduction to Linear Algebra and Tensors, Revised Edition download epub
Mathematics
Author: M. A. Akivis,V. V. Goldberg,Richard A. Silverman
ISBN: 0486635457
Category: Science & Math
Subcategory: Mathematics
Language: English
Publisher: Dover Publications; Revised edition (October 18, 2010)
Pages: 192 pages