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Introduction to Linear Algebra [Paperback]

Serge Lang

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Book Description

15 Mar 2012
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.
--This text refers to an out of print or unavailable edition of this title.

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Second Edition

S. Lang

Introduction to Linear Algebra

"Excellent! Rigorous yet straightforward, all answers included!"—Dr. J. Adam, Old Dominion University

--This text refers to an out of print or unavailable edition of this title.

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Amazon.com: 3.3 out of 5 stars  15 reviews
38 of 38 people found the following review helpful
5.0 out of 5 stars Excellent Introductory Text 2 Jan 2004
By Michael H. Borkowski - Published on Amazon.com
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.
17 of 17 people found the following review helpful
5.0 out of 5 stars A wonderful book and benchmark test for students 26 April 2000
By A Customer - Published on Amazon.com
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.
17 of 19 people found the following review helpful
4.0 out of 5 stars Good for upper level undergrads or motivated underclassmen 20 Sep 1999
By A Customer - Published on Amazon.com
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.
11 of 11 people found the following review helpful
4.0 out of 5 stars A nice introduction for students of pure mathematics. 13 Sep 2008
By N. F. Taussig - Published on Amazon.com
Serge Lang's Introduction to Linear Algebra provides a nice introduction to the subject. The text, which is designed for a one semester course for students who are taking or have completed multi-variable calculus, covers the basic theory and computational techniques. Since the emphasis is on proving theorems rather than the applications that are of interest to physical scientists, engineers, and economists, the text is best suited to pure mathematics students.

Topics are motivated, the theory is carefully developed, computational techniques are demonstrated through clearly written examples, and geometric interpretations of the algebra are discussed. The exposition is generally clear, but I occasionally had to turn to Blyth and Robertson's Basic Linear Algebra 2nd Edition or Friedberg, Insel, and Spence's Linear Algebra (4th Edition) for clarification when examples were lacking (notably in the section on eigenvalues and eigenvectors that precedes the introduction of the characteristic polynomial). Another caveat is that there are also numerous errors, including some in the answer key.

The exercises consist of computational problems, which require meticulous attention to detail, and proofs of results that extend the topics developed in the text. The exercises are organized thematically in order to teach concepts not covered in the body of the text. Some problems are reintroduced after additional material has been developed, so that you can solve them in new, more efficient, ways, thereby demonstrating the power of the new techniques that you are learning. Answers to most of the exercises are provided in an appendix, making the text suitable for self-study.

The text begins with a review of vectors. This material is drawn from Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics) and should be familiar to most readers. Next, Lang demonstrates how matrix algebra can be used to solve systems of linear equations. While the reader presumably learned how to solve systems of linear equations in high school (or even earlier), the discussion of homogeneous linear equations, row operations, and linear combinations provides the foundation for subsequent topics in the book.

The remainder of the book is devoted to finite-dimensional vector spaces. Once Lang introduces the basic definitions, he covers linear independence, the basis of a vector space, and dimension. This leads to a discussion of linear mappings, their representation by matrices, and how the kernel and image of these maps are related to the rank of the matrix of linear transformation. Lang discusses composition of mappings and inverse mappings before delving into scalar products, orthogonal bases, and bilinear maps. Lang then develops the theory of determinants and discusses how to apply them to solving systems of linear equations, finding the inverse of a matrix, and calculating areas and volumes. After introducing eigenvectors, eigenvalues, and the characteristic polynomial, Lang concludes the book with a discussion of the eigenvectors and eigenvalues of symmetric matrices that uses the earlier material on scalar products and orthogonality.

Much of this material is drawn from Lang's Linear Algebra (Undergraduate Texts in Mathematics), where it is treated in more depth. However, that text is written for students already familiar with basic matrix manipulations, so it does not discuss elementary matrices or Gaussian elimination. Understanding it also requires greater mathematical sophistication.

This text is limited in scope. If you are preparing to do graduate work in mathematics, you will need to read an additional text such as Lang's Linear Algebra, Friedberg, Insel, and Spence's Linear Algebra, Hoffman and Kunze's Linear Algebra (2nd Edition), Axler's Linear Algebra Done Right, or Blyth and Robertson's Further Linear Algebra. Of these, the one that is most suitable for self-study is Further Linear Algebra.

If you are interested in an introductory text that is suitable for self-study, you may wish to consider Blyth and Robertson's Basic Linear Algebra as an alternative to this one, as it includes abundant examples and answers to almost all the exercises.
7 of 7 people found the following review helpful
3.0 out of 5 stars Too Concise (3.5 Stars) 13 May 2006
By Madison, James - Published on Amazon.com
This book was used in our intro to linear algebra for math/physics majors, and it was decent, but not great. The book is very concise, so you can easily read it before your lecture and then again after lecture (most sections are about 5 pages). However, this also leads to some things not being hammered in, or not emphasized as much as they should be. Also, there are very few examples of actual problems, and the reader is often left having to do something concrete from the theory with no examples. This is a tad annoying, but is good in that it makes you think. However, there are a few errors in the answers in the back of the book. So there were a few times I did what I thought was the logical way to do a problem, and my answers were not the same as the books, but mine were correct. These errors in the book were very annoying, but the actual text, for the most part, had few errors. The style of writing is fine, and to the point.

Overall, the book is decent, but I am sure there are better books out there.
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