Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition [Hardcover]

Serge Lang (Author)

Book Description

ISBN-10: 0387962050 | ISBN-13: 978-0387962054 | Publication Date: December 19, 1985 | Edition: 2nd

This is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues. The book contains a large number of exercises, some of the routine computational type, while others are conceptual.

Editorial Reviews

Review

Second Edition S. Lang Introduction to Linear Algebra "Excellent! Rigorous yet straightforward, all answers included!"—Dr. J. Adam, Old Dominion University

Product Details

• Hardcover: 293 pages
• Publisher: Springer; 2nd edition (December 19, 1985)
• Language: English
• ISBN-10: 0387962050
• ISBN-13: 978-0387962054
• Product Dimensions: 9.3 x 6 x 0.8 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.2 out of 5 stars  See all reviews (13 customer reviews)
• Amazon Best Sellers Rank: #834,047 in Books (See Top 100 in Books)

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28 of 28 people found the following review helpful:

5.0 out of 5 stars Excellent Introductory Text, January 2, 2004

By 

Michael H. Borkowski - See all my reviews
(REAL NAME)   

This review is from: Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition (Hardcover)

This text is intended for a one semester introductory course in Linear Algebra at the sophomore level geared toward mathematics majors and motivated students. It was originally extracted from Lang "Linear Algebra," and is now in its second edition (a vast improvement over the first: Lang rarely does the increasingly popular token update).

The text takes a theoretical approach to the subject, and the only applications the reader can expect to see are to other interesting areas of mathematics. With the exception of the last chapter, these are left in the exercises, and Lang does not push them vary far.

The trend in most Linear Algebra texts at this level that attempt to appeal to a large audience (such as engineering students) is away from the Definition-Theorem-Proof approach and towards a less formal presentation based around ideas, discussions as proofs, and applications. I prefer the former approach, which Lang is very much in the tradition of, and believe that the way to teach students how to write rigorous and presentable proofs is by making them read and study them. In fact, I learned how to write proofs from studying this text and working all of Lang's well-chosen exercises.

"Introduction to Linear Algebra" starts at the basics with no prior assumptions on the material the reader knows (the Calculus is used only occasionally in the exercises): the first chapter is on points, vectors, and planes in the Euclidean space, R^n. After that is a chapter introducing matricies, inversion, systems of linear equations, and Gaussian elimination. While the book does spend adequate time on how to perform Gaussian elimination and matrix inversion, it also gives all the proofs that these methods work.

The bulk of the theoretical material comes in Chapters III through V, which respectively present the theories of vector spaces, linear mappings, and composite and inverse mappings. The approach is rigorous, but by no means inaccessible. As is necessary in a course like this, time is spent on establishing clear and solid proofs of basic results that will be treated as almost trivial ("you can show it on your homework to convince yourselves") in more advanced classes - c.f. Lang's "Undergraduate Algebra."

The next two chapters cover scalar products and determinants, and have a somewhat more computational feel to them. There is much theory in the sections on scalar products, but a big focus is also the Gramm-Schmidt method for finding an orthonormal basis. Many of the determinant proofs are in the 2 x 2 and 3 x 3 case to avoid bringing in the full formalism and notation of determinants in general.

The text concludes with what is its most difficult chapter, the one on eigenvectors and eigenvalues. It is the most, however, for applications to physics, and interest applications comprise the last half of the chapter.

If you are ordering this text used, I recommend you take care to find the second edition. The first edition was significantly shorter and covered less material.

This is an introductory text, and not for learning the material that would be included in a second course or part of the algebra sequence at the junior/senior level. For those purposes, I recommend Lang's "Linear Algebra." Portions bear strong (often exact) resemblance to the book at present consideration, but the most basic material is missing and much advanced material is included.

In conclusion, I highly recommend this text for a motivated student who wants a first exposure to Linear Algebra. The text isn't always easy reading, and parts may be a tough climb for readers without much exposure to this type of reading. The experience, however, is well worth it; in mathematics, one really only learns as much as one sweats, so to speak.

14 of 14 people found the following review helpful:

5.0 out of 5 stars A wonderful book and benchmark test for students, April 26, 2000

By A Customer

This review is from: Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition (Hardcover)

This is a wonderful book for freshmen/sophomores. Being a senior now, it's easier to evaluate the quality of the text and judge it's worth compared to other books. I really don't think there's a better book on linear algebra at this level. Everything in here is well motivated, organized and as rigorous as possible for an intro book. That's not to say that there's not room for improvement as far as motivation goes, but what he has certainly suffices. Even if you don't get everything in here on your first pass, this book provides a good benchmark test - if you can get through it in good shape, then you are probably well prepared to begin upper level work. If you can't, then you should probably try again before attempting a serious course in, say, group theory or topology. Linear algebra provides the ideal subject matter with which to introduce the student to rigorous proof techniques, because it has so many easily visualized yet useful examples. So if you can't follow the proofs here, don't expect to follow the proofs in a more abstract course. If there's any other book that I might use in this one's place, it would actually be Lang's "Linear Algebra," which I find to be more cohesive and motivated, although more difficult.

15 of 17 people found the following review helpful:

4.0 out of 5 stars Good for upper level undergrads or motivated underclassmen, September 20, 1999

By A Customer

This review is from: Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition (Hardcover)

This text is well written and is motivated by theory. Better suited as a supplemental text, as opposed to a required course text. As for the self acclaimed "smart" reviewer from Irvine, A grades in mathematics do not mean that you have mastered the subject. The school which you attend also affects your grades. Not to mention, linear algebra is usually a bridge to higher mathematics, grades in calculus, and diff equs don't really matter. If you are having trouble with this text or the course associated with it, chances are that you will have a very, very hard time in more mathematical courses such as abstract algebra and classical analysis. Calculus 1, 2, 3, and diff equs are just applications of mathematical theory. It is doubtful that after this sequence that the student even knows the definition of a limit of a function in a single variable which is ironic, for what is calculus but the study of limits. For example, the derivative is the limit of the difference quotient, the Riemann integral is the limit of Riemann sums, etc...

The point is that linear algebra, at the theoretical level, is a bridge to higher mathematics. This is a good text to use in order to cross that bridge. Serge Lang is a great mathematician, he was recently given an award from the AMS for his achievements in writing textbooks. My favorite linear algebra text is 'Linear Algebra' by Hoffman/Kunze. For people who lack in mathematical ability and wish for a more applied introduction to linear algebra, 'Linear Algebra and its Applications' by Gilbert Strang. If the reader from Irvine or anyone having difficulty with this beautiful subject wishes for a even simpler text, there is 'Elementary Linear Algebra' by Bernard Kolman.

Amongst other great works by Serge Lang, I believe that 'Algebra' is a classic which should be in the library of any mathematics student and professor.