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Linear Algebra [Hardcover]

Serge Lang
5.0 out of 5 stars  See all reviews (1 customer review)

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Product details

  • Hardcover: 296 pages
  • Publisher: Springer; 3rd edition (2004)
  • Language: English
  • ISBN-10: 0387964126
  • ISBN-13: 978-0387964126
  • Product Dimensions: 23.6 x 16.3 x 2.5 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 1,492,038 in Books (See Top 100 in Books)

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Product Description

Offers an exposition of the basic theory of vector spaces. This book explains the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form.

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Most Helpful Customer Reviews
3 of 3 people found the following review helpful
By A Customer
An excellent book, he takes beginners right up to advanced level. Very useful when taking an advanced course in linear algebra, if you haven't really broached the topic before. Far too expensive but I wouldn't let that put you off. There aren't enough worked examples, but if you buy this in conjunction with perhaps "Linear Algebra" by Seymour Lipschutz in the Shaum's outlines series, you will be fine.
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Most Helpful Customer Reviews on (beta) 4.2 out of 5 stars  9 reviews
28 of 32 people found the following review helpful
5.0 out of 5 stars A very concise and rigorous book 27 April 2000
By A Customer - Published on
It is the rare sort of book that makes an excellent introduction for the student and useful reference for the graduate, and that is why I recommend this book unreservedly. Its conciseness may leave some students adrift, but I think it serves to enhance the book's structural coherence. And while the given proofs are usually terse, I think that makes the book a wonderful way to measure one's prepardness for upper level studies. Linear algebra is the ideal subject with which to familiarize students with rigorous proof techniques because it has so many easily visualized yet useful examples. If one cannot understand Lang's proofs here, one is probably not ready to tackle an upper level course in, say, algebra or topology. This book therefore serves as a proving grounds, so to speak (and excuse the pun!), for one's mathematical ability. Survive the test and you'll probably do well in your upper level studies, otherwise you should practice some more and try again.
7 of 7 people found the following review helpful
5.0 out of 5 stars Strong concise book for linear algebra over the Complex numbers 20 Aug 2009
By MrDNA - Published on
Lang's Linear Algebra is one of my favorite undergraduate math books. The style is concise and clear, and the approach is rather rigorous. I found the chapters on polynomials particularly interesting.

Minor complaints: (1) I would rather Lang have done the book over general (e.g. finite) fields rather than sticking with subfields of the complex numbers. That way it would be more clear which results truly rely on results in Complex analysis and which rely only on the fact that the every n-degree polynomial over the complex numbers has n roots. (2) A couple more examples involving function vector spaces would have been interesting. Although all-in-all he strikes a great balance for as short a book as this is.
18 of 23 people found the following review helpful
5.0 out of 5 stars Solid piece of work 26 Oct 2004
By J. C. Verheul - Published on
Like in other math books by Lang, the theory of Linear Algebra is presented in an axiomatic way, the best way of presenting since The Elements of Euclid. The way in which the theory is presented adds to the beauty. I have read this book as a refresher for Linear Algebra, about 20 years after the completion of a master's degree in an exact science. For me the level was perfect. If you have no experience with Linear Algebra beyond high school, you must first read "Introduction to Linear Algebra" by Lang or some other introductory course. The book under review does not talk about basics like Gauss-elimination. I have seen remarkably few typos. Some cross-references to theorems in other chapters were wrong, though. In all: a very good book and well worth the money.
8 of 10 people found the following review helpful
4.0 out of 5 stars Difficult for undergrads 10 Aug 2006
By TMS - Published on
I think the term "undergraduate" is a bit misleading. I think you would have had to have at least one course in linear algebra and abtract algebra to truly appreciate this book. I read it over a summer (as a master's student who lacked any coursework in linear algebra) - kind of as an independent project, and I found it to be very easy to understand. Then again, I had just taken abstract algebra. There were a couple parts that I found challenging though. I love it when he says that the proof or rest of the proof is "trivial" and unnecessary to write. I have heard he does this in many of his books. Overall great book if you have some background.
17 of 24 people found the following review helpful
4.0 out of 5 stars This Book Is Almost Excellent 25 April 2003
By Khalifa Alhazaa - Published on
I had this book as the text for my second course in Abstract Algebra, having already taken some elementary Linear Algebra course. I might argue that this is not the best subject for such a course, yet this is very irrelevant here.
All through the class I struggled to understand concepts. I did not. That was not due to the book since I did not even bother opening it. After finishing the course, I realized that the material of this book is of the most importance to anyone planning on continuing his/her grad degree in math, so I decided to read the book. The mission was accomplished in a matter of a couple of weeks.
I do not claim that this is the easiest book to understand the material. In fact, Lang's books are remarked for their dryness. Motivation is almost nonextant. If you, however, have a fairly good background in Linear Algebra (something like the material of Anton's "Elementary Linear Algebra" or the like) and Abstract Algebra (an excellent introduction can be sought in Herstein's "Abstract Algebra") you would much benefit from this book.
The book is a very good book for a second course in linear algebra, that is, it is not a good book for those who had no experience with matrix theory before. The reason is that the book does not mention anything about Gaussian elimination and treats the solutions of n equations in m unknowns using dimension theorems, which is not the standard way of proving existence of such solutions. One more thing is that it does not talk about elementary matrices (one can interpret column or row operations by multiplication of elementary matrices to the right or left). I am not saying the book
is bad, I am saying it is not the right book for a beginner.
The book introduces the basic notions of vector spaces, linear mappings, matrices scalar products, determinants, and eigenvalues and spaces. It then moves to unitary, symmetric, and Hermitian operators and explores their Eigenvalues. Polynomials have a whole chapter followed by triangulation of a linear map. The book concludes with applications of Linear algebra to convex geometry.
I might disagree with the definition of the determinant the author offers, but I would have to admit that his approach is the traditional one.
The subjects of the books must be mastered (or at least absorbed) by anyone who wants to go to analysis (Functional analysis to be precise), Algebra, Geometry, and Differential Equations. To ensure this you should do almost all the exercises of the book since they are so excellent and help a lot in understanding the material presented.
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