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1. Linear Algebra (Graduate Texts
2. Multilinear Algebra
3. Linear Algebra: An Introductory
4. Multilinear Algebra (Universitext)
5. Advanced Linear Algebra (Graduate
6. Applied Linear Algebra and Matrix
7. Division Algebras:: Octonions
8. Linear Algebra (Undergraduate
9. Further Linear Algebra
10. Algebras, Rings and Modules: Volume
11. Introduction to Linear Algebra
12. Linear Algebra Through Geometry
13. Multilinear Algebra (Algebra,
14. Abstract Linear Algebra (Universitext)
15. Linear Algebra
16. Analysis of Dirac Systems and
17. Numerical Linear Algebra for Applications
18. Finite dimensional multilinear
19. On the Teaching of Linear Algebra
20. Linear Algebra (Undergraduate

1. Linear Algebra (Graduate Texts in Mathematics) (v. 23)
by Werner H. Greub
Hardcover: 476 Pages (1975-06-30)
list price: US$99.00 -- used & new: US$74.11
(price subject to change: see help)
Asin: 0387901108
Average Customer Review: 4.0 out of 5 stars
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This textbook gives a detailed and comprehensive presentation of the linear algebra based on axiomatic treatment of linear spaces. The author maintains a good balance between modern algebraic interests and traditional linear algebra. Several chapters have been substantially rewritten for clarity of exposition, although their basic content is unchanged. A considerable number of exer- cises covering new material has also been added. ... Read more

Customer Reviews (1)

4-0 out of 5 stars rigorous discourse
There are many elementary texts on linear algebra. Often at a level well suited to a student in science or engineering that needs a pragmatic approach to the subject as a tool that is a means to another end. But for maths majors, those books might come up lacking.

In contrast, Greub offers a rigourous discourse. In principle, starting from scratch. But realistically, you do need a pre-existing and strong background in some of the ideas. For example, eigenvectors and eigenvalues are discussed, along with the corresponding characteristic equation. The importance of this idea might not be sufficiently emphasised by the text. Whereas your physics classes should already have made you aware of its seminal nature.

The book is a good sequel to Peter Halmos' classic Finite-Dimensional Vector Spaces. It expands greatly on ideas that Halmos only touched on. ... Read more

2. Multilinear Algebra
by D. G. Northcott
Paperback: 216 Pages (2009-01-18)
list price: US$43.00 -- used & new: US$31.00
(price subject to change: see help)
Asin: 0521090601
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Multilinear algebra has important applications in many different areas of mathematics but is usually learned in a rather haphazard fashion. The aim of this book is to provide a readable and systematic account of multilinear algebra at a level suitable for graduate students. Professor Northcott gives a thorough treatment of topics such as tensor, exterior, Grassmann, Hopf and co-algebras and ends each chapter with a section entitled 'Comments and Exercises'. The comments contain convenient summaries and discussion of the content whilst the exercises provide an opportunity to test understanding and add extra material. Complete solutions are provided for those exercises that are particularly important or used later in the book. The volume as a whole is based on advanced lectures given by the author at the University of Sheffield. ... Read more

3. Linear Algebra: An Introductory Approach (Undergraduate Texts in Mathematics)
by Charles W. Curtis
Hardcover: 364 Pages (1984)
list price: US$69.95 -- used & new: US$35.00
(price subject to change: see help)
Asin: 0387909923
Average Customer Review: 4.0 out of 5 stars
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This revised and updated fourth edition designed for upper division courses in linear algebra includes the basic results on vector spaces over fields, determinants, the theory of a single linear transformation, and inner product spaces. While it does not presuppose an earlier course, many connections between linear algebra and calculus are worked into the discussion. A special feature is the inclusion of sections devoted to applications of linear algebra, which can either be part of a course, or used for independent study, and new to this edition is a section on analytic methods in matrix theory, with applications to Markov chains in probability theory. Proofs of all the main theorems are included, and are presented on an equal footing with methods for solving numerical problems. Worked examples are integrated into almost every section, to bring out the meaning of the theorems, and illustrate techniques for solving problems. Many numerical exercises make use of all the ideas, and develop computational skills, while exercises of a theoretical nature provide opportunities for students to discover for themselves. ... Read more

Customer Reviews (8)

3-0 out of 5 stars Terse text but easy exercises
As other readers have noted, this is your standard, terse, theorem-lemma-proof type math text.If you are approaching this text with your only mathematical background being a popular treatment of the calculus like those of Stewart or Larson, the style takes a lot of getting used to.Though a lot of effort is placed into making the text short and packed with information, it is impossible to sit down with this text (or any other upper-division math text, for that matter) and read passively.Curtis leaves out many details in the proofs and examples, asking the reader to follow the ordinary, though annoying, convention of theorem-lemma-proof texts and provide the details him(yes, him)self.

In my linear algebra class, we covered the equivalent of the first six chapters, though after the determinants chapter, we stopped following the ordering and pace of the text, and therefore, it's hard to tell whether the later chapters make an optimal teaching or self-study device.The first four chapters (up to determinants), however, take a theoretical approach to linear algebra with a few scattershot references to the geometric intuitions that it formalizes.

My biggest complaint is that in some points of the text, Curtis seems unsure of whether to continue in an informal or an abstract manner.Many of the exercises are numerical problems of the type you see in a non-proof linear algebra class focused on applications, and when Curtis gets to systems of linear equations, he seems confused about whether he is writing a book for a proof-based or a numerical class.As long as you are familiar with induction and proof by contradiction, the exercises that require proofs are fairly easy.Our professor constantly supplemented them with problems that were far more difficult than anything in Curtis.

5-0 out of 5 stars Good Book for Introductory Linear Algebra
This is a fantastic resource for basic linear algebra. It covers the topics from a rather theoretical perspective, which lacks many examples or practical applications, but presents the theory better than most alternate books. The book does include some applications/advanced theory moving past the basic topics of solutions of systems, vector spaces, linear transformations, determinants, and orthogonality, which includes solving differential equations and basic symmetry theory. The book is short and sweet, which is good if you can handle more advanced math textbooks, but not good for those looking for detailed explanation (some of the proofs of basic concepts are even left up to the reader when appropriate). This book is better for those already familiar with the subject matter or those looking for a challenge.

4-0 out of 5 stars Used for Self Study
I used this book for an independent study of linear algebra in undergrad.I'd had no prior exposure to the field, but had taken very good real analysis and abstract algebra courses prior.My self-study was quite effective, which is a testament to the book, since it was essentially the only material I had.(There were a few typos I remember seeing, as well as some unclear formulas/notation here and there, hence the four stars, but nothing serious).

5-0 out of 5 stars Beautiful Work
I used this book for a first course in linear algebra at Ohio State University and I could not have been more satisfied.It is a challenging book to use, but very rewarding for those who put the time into it.

3-0 out of 5 stars OK, but could be better.
The book sometimes eliminates using some symbols when the meaning is clear from the context; this is OK only if you either have prior understanding of the material presented or easy access to an expert. ... Read more

4. Multilinear Algebra (Universitext)
by Werner Greub
 Paperback: 294 Pages (1978-08-24)
list price: US$32.50
Isbn: 0387902848
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5. Advanced Linear Algebra (Graduate Texts in Mathematics)
by Steven Roman
Paperback: 526 Pages (2010-11-02)
list price: US$74.95 -- used & new: US$60.14
(price subject to change: see help)
Asin: 1441924981
Average Customer Review: 5.0 out of 5 stars
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This graduate level textbook covers an especially broad range of topics. The book first offers a careful discussion of the basics of linear algebra. It then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators. The new edition has been revised and contains a chapter on the QR decomposition, singular values and pseudoinverses, and a chapter on convexity, separation and positive solutions to linear systems.

... Read more

Customer Reviews (3)

5-0 out of 5 stars The true "Linear ALGEBRA".
This is linear algebra as rigorous and beautiful as it can be.

The development of the presented concepts is in terms of advanced abstract algebra; particularly
rings and modules. However, no prior knowledge of abstract algebra is required to study the book,
and this is the very element that makes it a masterpiece. Readers unfamiliar though with basic
abstract algebra should expect to progress quite slowly throughout the book, given that they
show devotion and enough faith to let their minds be guided by the author's educating pen.

Diligent study of the book leads to beautiful insights being emerged that are far from trivial. For
this reason, it is highly recommended that the prospective reader possesses a good level of mathematical
maturity and is already familiar to a sufficient degree with linear algebra and matrix analysis, e.g. at
the level of C. Meyer's "Matrix Analysis and Applied Linear Algebra", which is an excellent first study
of the topic.

"Advanced Linear Algebra" is a book that addresses those who love linear algebra (or algebra) and are
serious about mastering the pertaining concepts. The book is invaluable for combining rigor, depth of
exposition, excellent notation and educational character for a topic of immense importance that pervades
almost every aspect of modern quantitative sciences.

5-0 out of 5 stars Outstanding clarity; this is a very well-written book
Mathematics books are often considerably more difficult to read than their authors prepare their audiences to believe; this book is a happy exception.It is written for an audience of readers at a specific place in their studies (ones who know linear algebra but want to take their understanding of it to a deeper level), and it reaches this audience very well.The emphasis of this book is on linear algebra in abstract mathematics; it is less useful for people interested in numerical linear algebra.

As the name suggests, this book requires a fair amount of background.The introductory chapter moves very fast, but is thorough, and exciting to read.The rest of the book presents advanced topics at a more leisurely pace, while still remaining fairly concise.Some difficult concepts, such as the universal property, are introduced several times at several different places in the book, so that someone working through the book will be more familiar with them when it is finally necessary to understand them on a deeper level.

I find the material on modules outstanding; the author explores the analogies between modules and vector spaces, rigorously exploring which analogies hold, and giving examples of cases in which other analogies fail.The presentation of modules in this book differs greatly from that encountered in most abstract algebra texts: while most books focus on modules' similarities to rings and applications in commutative algebra, this text focuses on their similarities to vector spaces and applications to the study linear operators on vector spaces.

One should not be scared by the word "advanced" in the book's title.Although the book covers advanced topics, it is very clear.When proofs are omitted, it is usually because they are very easy for the reader to supply.The exercises are very valuable (some are critical for understanding the material), but they're not diabolically difficult.

I think this book would make an outstanding textbook for an introductory graduate-level course in linear algebra, or perhaps a senior-level undergraduate course for students with a strong background.It is also very well-suited to self-study.A student with prior background in abstract algebra (group theory, ring theory, etc.) will find this book much more manageable than a student who has not covered such material.People wanting a more introductory text might want to look to the book by Axler, or the old classic by Shilov.

5-0 out of 5 stars A real treasure
Linear algebra is crucial to anyone in a mathematical or technical field. To the pure or applied mathematician, it is the bread and butter -- a lot of fundamental theorems (even in quite advanced fields like algebraic geometry) ultimately come down to a calculation using linear algebra.

In any case, this book is brilliant for the moderately advanced student who knows the basics (maybe sketchily) and wants an extremely comprehensive, rigorous, and coherent review and reordering of his or her linear algebra knowledge. I knew most of the topics in this book in a superficial way, but reading it is quite fulfilling because it all comes together at once.

The choice of topics and the angles from which they are presented is extremely strong. The Jordan and Rational Canonical Forms get a full and rigorous treatment. Unlike many linear algebra books, which use some ugly matrix-related kludge in the proofs of the classification theorems, this book does these topics from the algebraic perspective (i.e., as decompositions of modules over principal ideal domains). Inner product spaces are done in their own substantial chunk of the book, where all the essential ideas are developed abstractly and well. Sometimes linear algebra books focus too much on particular examples of inner product spaces or resort to "magical" proofs of important inequalities. This book takes care to build up important lemmas so that big results fall out "naturally". It is by far the best abstract treatment of inner products that I have read (although it should be supplemented by a knowledge of some of the standard examples, which can be found in a typical introductory textbook).

The proofs are the most elegant possible, with no ugliness or nonsense. The notation is a gem, without confusing mixes of superscripts and subscripts and nonstandard choices. The exposition is at just the right level (for me at least) -- the steps in proofs that are left as exercises are all reasonable and straightforward, and all the details that are subtle or interesting are filled in, discussed, and emphasized.

I have been looking for a beautiful book on linear algebra of this sort for a long time, and am delighted to have finally found it.
... Read more

6. Applied Linear Algebra and Matrix Analysis (Undergraduate Texts in Mathematics)
by Thomas S. Shores
Hardcover: 388 Pages (2006-12-06)
list price: US$79.95 -- used & new: US$53.03
(price subject to change: see help)
Asin: 0387331948
Average Customer Review: 4.0 out of 5 stars
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This new book offers a fresh approach to matrix and linear algebra by providing a balanced blend of applications, theory, and computation, while highlighting their interdependence. Intended for a one-semester course, Applied Linear Algebra and Matrix Analysis places special emphasis on linear algebra as an experimental science, with numerous examples, computer exercises, and projects. While the flavor is heavily computational and experimental, the text is independent of specific hardware or software platforms. Throughout the book, significant motivating examples are woven into the text, and each section ends with a set of exercises.

... Read more

Customer Reviews (2)

3-0 out of 5 stars good, could be better
I have adopted this book as one of the texts for my introductory course on applied math. It's well written and has a diverse set of good examples drawn from everything from computer graphics to sports betting. It's fun to read, and in general, good.

It is significantly better than Olver and Shakiban's book of similar title and scope, and a heck of a lot cheaper. In a course like mine where I use several specialized books rather than one of the massive tomes that try (and fail) to cover all of applied math, low price is a factor for me.

Negatives include: too few theory problems, a fair number of typos, relegation of the LU factorization to a problem set. On the whole, though, it's not bad.

5-0 out of 5 stars Stimulating introduction to Linear Algebra
This is a very well written book. The author brings the 'topic to life' by showing how the linear algebra can be used in an applied setting. Also the graphics enable the reader to understand the material from a geometric perspective, as opposed to merely looking at terse algebraic equations.

Finally, the hints and answers section at the back of the book are very good for self study.

Highly recommended! ... Read more

7. Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics (Mathematics and Its Applications)
by G.M. Dixon
Paperback: 248 Pages (2010-11-02)
list price: US$115.00 -- used & new: US$91.73
(price subject to change: see help)
Asin: 1441947469
Average Customer Review: 5.0 out of 5 stars
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The four real division algebras (reals, complexes, quaternionsand octonions) are the most obvious signposts to a rich and intricaterealm of select and beautiful mathematical structures. Using the newtool of adjoint division algebras, with respect to which the divisionalgebras themselves appear in the role of spinor spaces, some of thesestructures are developed, including parallelizable spheres,exceptional Lie groups, and triality. In the case of triality the useof adjoint octonions greatly simplifies its investigation. Motivatingthis work, however, is a strong conviction that the design of ourphysical reality arises from this select mathematical realm. Acompelling case for that conviction is presented, a derivation of thestandard model of leptons and quarks.
The book will be of particular interest to particle and high energytheorists, and to applied mathematicians.
... Read more

Customer Reviews (2)

5-0 out of 5 stars I am waiting for Dixon 's Octonians...
I have not yet received my command. P. MERAT

5-0 out of 5 stars Mathematics behind physics
This is an excellent book for those who want to study Hamilton's quaternions, and other algebraic structures, used in modern physics. Dixon believes that octonions and triality of Spin(8) are essential in understanding particle physics. This clear exposition contains many ideas which have gone unnoticed from other researchers. The book is a treasure trove for mathematical physicists. The author also compares the Cayley algebra of octonions to other algebraic systems used in physics: matrices and Clifford algebras, in particular the Dirac algebra. ... Read more

8. Linear Algebra (Undergraduate Texts in Mathematics)
by Serge Lang
Paperback: 296 Pages (2010-11-02)
list price: US$64.95 -- used & new: US$52.25
(price subject to change: see help)
Asin: 1441930817
Average Customer Review: 4.0 out of 5 stars
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This book begins with an exposition of the basic theory of vector spaces and proceeds to explain 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. Material in this new edition has been rewritten and reorganized and new exercises have been added.

... Read more

Customer Reviews (7)

5-0 out of 5 stars Strong concise book for linear algebra over the Complex numbers
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.

4-0 out of 5 stars Difficult for undergrads
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.

5-0 out of 5 stars Solid piece of work
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.

5-0 out of 5 stars Serge Lang is a Very Gifted Expositor
Serge Lang is a very gifted expositor. I've read the reviews saying that his books are notorious for their "dryness". At least as concerns this book - that couldn't be less true.

This book is not only methodical and well written, it is a joy. Every section is a well rounded presentation: Lang clearly and effectively introduces new concepts and patiently develops even the most basic results. But Lang achieves much more: his illuminating examples are stepping stones to a more abstract understanding.

Enjoying this book is like enjoying anything of high quality and craftsmanship. Admittedly, that is not always for everyone to enjoy.

4-0 out of 5 stars This Book Is Almost Excellent
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. ... Read more

9. Further Linear Algebra
by Thomas S. Blyth, Edmund F. Robertson
Paperback: 235 Pages (2001-11-09)
list price: US$59.95 -- used & new: US$29.85
(price subject to change: see help)
Asin: 1852334258
Average Customer Review: 4.0 out of 5 stars
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Further Linear Algebra is a natural sequel to the authors' highly acclaimed Basic Linear Algebra (ISBN 3-540-76122-5). The more advanced topics covered here take the reader to the very heart of the subject, and include inner product spaces, direct sums of subspaces, the primary decomposition theorem and various canonical forms for matrices. Furthermore, the authors provide a brief introduction to the use of MAPLE in linear algebra calculations, and biographical profiles of eminent mathematicians associated with the subject. ... Read more

Customer Reviews (2)

3-0 out of 5 stars Buy With a Bottle of Glue...
Or you may lose a few pages.
The first day of use a pair of sheets came undone.I glued them in and now another set is coming out.The mathematical material looks well done and is nice pedagogical work.I would have given the book 4 or 5 stars for that, but the physical packaging--a responsibility of the publisher--is terrible.Why didn't they sew in the pages of a math text, which always can be counted on to receive heavy use?The first day of use and it begins to fall apart, too!Glue alone is not enough; I've owned cheap paperback novels which were bound more durably.Shame on Springer, a venerable science publisher!

5-0 out of 5 stars Fascinating, rigorous and fun
I used this book as part of a Linear Algebra 2 course and found it to be clear and enjoyable to learn from.The chapter on the history of the major pioneers was particularly delightful.This book contains more than most Linear Algebra 2 courses cover (at least more than we did) so I wouldn't necessarily recommend it as THE book to buy to learn Linear Algebra.As its name indicates, it is more of an enrichment book in many ways.I think that this book, like many put out by Springer, helps flesh out an otherwise daunting subject, showing part of its mathematical beauty that is visible on the undergraduate level. ... Read more

10. Algebras, Rings and Modules: Volume 2 (Mathematics and Its Applications)
by Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko
Paperback: 400 Pages (2010-11-02)
list price: US$169.00 -- used & new: US$169.00
(price subject to change: see help)
Asin: 9048172926
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This second volume of this text covers the classical aspects of the theory of groups and their representations. It also offers a general introduction to the modern theory of representations including the representations of quivers and finite partially ordered sets and their applications to finite dimensional algebras. It reviews key recent developments in the theory of special ring classes including Frobenius, quasi-Frobenius, and others.

... Read more

11. Introduction to Linear Algebra (Undergraduate Texts in Mathematics) 2nd edition
by Serge Lang
Hardcover: 293 Pages (1985-12-19)
list price: US$74.95 -- used & new: US$47.20
(price subject to change: see help)
Asin: 0387962050
Average Customer Review: 3.5 out of 5 stars
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This book 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, and others are conceptual. ... Read more

Customer Reviews (11)

1-0 out of 5 stars Missing Pages
I started using this book as it was the assigned for a course. However, it's mid-semester, and I realized my book is missing pages! I'm pretty sure it's a binding issue....

I'm very disappointed, and it's a huge annoyance to pay for a textbook that is missing pages. I wanted to study for an upcoming an exam, but will have to borrow a friend's book in the meantime. Do not buy this book, get the hard-copy book if possible.

5-0 out of 5 stars A good math textbook
This book is easy to understand, and very concise.There are a few printing errors, but I'm pleased with it.The answers are in the back of the book, but sadly no hints or detailed solutions.My professor chose to provide those as additional course materials, which I have found helpful.

Overall, a better than average math text.

4-0 out of 5 stars A nice introduction for students of pure mathematics.
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 whoare 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.

2-0 out of 5 stars Marginal Text
I used this book for a college-level linear algebra course and found it marginal. Although the author is clear, he skips steps and does not provide ANY answers to exercises in the back of the book. Sometimes you can find the answers to exercises in proofs contained in subsequent chapters. You really need a good professor (which I thankfully had) to get the most out of this course and book.

3-0 out of 5 stars Too Concise (3.5 Stars)
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. ... Read more

12. Linear Algebra Through Geometry (Undergraduate Texts in Mathematics)
by Thomas Banchoff, John Wermer
Hardcover: 324 Pages (1991-11-25)
list price: US$74.95 -- used & new: US$46.01
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Asin: 0387975861
Average Customer Review: 4.0 out of 5 stars
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Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry. ... Read more

Customer Reviews (2)

3-0 out of 5 stars Unreadable
This is a very poorly written and highly opaque book. For example, chapter 2.7 (sic) is called "Classification of Conic Sections", but conic sections are not even mentioned until five pages into this 12-page chapter. Apparently, Banchoff and Wermer finds it appropriate that a reader who wants to learn about the classification of conic sections should be forced to wade through five pages of technicalities, including things like six exercises on computing powers of various matrices, without the slightest indication of what this has to do with the classification of conic sections. Another instance, one among many, where Banchoff and Wermer demonstrate their commitment to technical nonsense and aversion to broad understanding is chapter 2.5 on determinants. "The quantity ad-bc is called the determinant", etc., and then the opening paragraph ends "We shall see that the determinant gives us further information about the behaviour of A" (p. 61), and there follows a devastatingly boring and longwinded five-page discussion on the orientation of an ordered pair of vectors, which is a technicality that could and should be dismissed in one paragraph. Only after this do we see that determinants have to do with area, which is the defining geometric property of the determinant, and of enormous importance. Why not do it the other way around? Why not say straight away that determinants are areas and then deal with the technical matter of orientation and signed areas at the end of the section, instead of keeping the reader in the dark with the secretive and mysterious proclamation that the determinant "gives us further information about the behaviour of A"? Banchoff and Wermer also adhere to a most unfortunate dichotomy between two, three, and n dimensions, which makes the book clumsily structured and repetitive. So, for example, there are three sections called "Linear transformations and matrices" (2.2, 3.2, 4.2), which all do basically the same thing, but our authors still pretend in 4.2 that we know nothing about linear transformations and matrices: "The symbol [(a_ij)] is called the matrix of the transformation ... Any transformation which can be written in this form is called a linear transformation", etc. (p. 213).

5-0 out of 5 stars A wonderful introduction
A crystal clear book.It shows the geometrical significance of topics like eigenvalues/eigenvectors without losing the intuition in the formalism, like so many other books on Linear Algebra.The book would be particularly suitable for scientists, though prospective mathematicians would appreciate the "whys" too.

I only wish there were follow-on, more advanced, volumes! ... Read more

13. Multilinear Algebra (Algebra, Logic and Applications)
by Russell Merris
Hardcover: 340 Pages (1997-08-20)
list price: US$139.95 -- used & new: US$122.92
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Asin: 9056990780
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The prototypical multilinear operation is multiplication. Indeed, every multilinear mapping can be factored through a tensor product. Apart fromits intrinsic interest, the tensor product is of fundamental importance in a variety of disciplines, ranging from matrix inequalities and grouprepresentation theory, to the combinatorics of symmetric functions, and all these subjects appear in this book.
Another attraction of multilinear algebra lies in its power to unify such seemingly diverse topics. This is done in the final chapter by means of therational representations of the full linear group. Arising as characters of these representations, the classical Schur polynomials are one of the keys to unification.
Prerequisites for the book are minimized by self-contained introductions in the early chapters. Throughout the text, some of the easier proofs areleft to the exercises, and some of the more difficult ones to the references. ... Read more

14. Abstract Linear Algebra (Universitext)
by Morton L. Curtis
Paperback: 184 Pages (1990-06-25)
list price: US$59.95 -- used & new: US$44.35
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Asin: 0387972633
Average Customer Review: 5.0 out of 5 stars
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Beginning from scratch and developing the standard topics of Linear Algebra, this book is intended as a text for a first course on the subject. The goal to which this work leads is the Theorem of Hurwitz - that the only normed algebras over the real numbers are the real numbers, the complex numbers, the quaternions, and the octonions. Unique in presenting this material at an elementary level, the book stresses the complete logical development of the subject and will provide a bavuable reference for mathematicians in general. ... Read more

Customer Reviews (1)

5-0 out of 5 stars An illuminated, highly concentrated review of Linear Algebra
From a physics undergrad: This is one of those wonderful, highly concentrated books that deliver extensive Wisdom in only a few pages of hardcore, to-the-point Theory, without simplifying. If you enjoy being told things in a frank manner and without excessive preambles, even if this means slower reading, this book is for you. The theory is built from scratch, but a lack of general abstract mathematical culture might prove problematic. Not recommended for the unprepared or those greedily seeking immediate applications. ... Read more

15. Linear Algebra
by Harold M. Edwards
Paperback: 184 Pages (2004-10-15)
list price: US$74.95 -- used & new: US$55.14
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Asin: 0817643702
Average Customer Review: 5.0 out of 5 stars
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In his new undergraduate textbook, Harold M. Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra. Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century.

Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience.

Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject. Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject.

... Read more

Customer Reviews (1)

5-0 out of 5 stars Highly recommended! All solutions included
So you may know who proved the Fermat's Last Theorem! The author, if I remember correctly, wrote some sensational papers and books on the subject as well. He is a great mathematician. His style is clear and interesting. I like his works a lot. Well, maybe I am biased. Who is not?! Anyhow, this linear algebra book is a great text. Clear, concise, to the point, stimulating, very engaging! Moreover, it contains solutions to all the problems in the text--the only linear algebra text that offers complete solutions! Your professor probably won't use it as a text but you may want to have it since you can learn it yourself even you miss some points in class... ... Read more

16. Analysis of Dirac Systems and Computational Algebra (Progress in Mathematical Physics)
by Fabrizio Colombo, Irene Sabadini, Frank Sommen, Daniele C. Struppa
Hardcover: 336 Pages (2004-09-23)
list price: US$104.00 -- used & new: US$25.00
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Asin: 0817642552
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Since their inception, Clifford algebras have been used as a mathematical tool for the description of physical phenomena. However, Clifford algebras have become an increasingly rich area of research with a significant number of important applications not only to mathematical physics, but to numerical analysis, harmonic analysis, and computer science as well. This book details: Systems of first-order linear differential equations that arise naturally in the setting of Clifford algebras; some well-known systems that play a role in electromagnetism and quantum mechanics, and others that figure prominently in harmonic analysis.

The reader is also introduced to new classes of special first order differential equations that have never been studied before, with applications ranging from multiparticle theory to finite geometry.These powerful new tools are applied to the study of Dirac and Dirac-like systems, Maxwell and Proca systems, and higher spin systems such as the Rarita-Schwinger system.

Given the richness of new ideas and research problems, this work should serve advanced undergraduates and graduate students in mathematics, physics and computer science, as well as researchers in Clifford analysis, differential equations, Grobner bases, computer algebra, quantum field theory and microlocal analysis. Only a basic course in complex analysis is required; the fundamentals of Clifford analysis are provided.

... Read more

17. Numerical Linear Algebra for Applications in Statistics
by James E. Gentle
Hardcover: 221 Pages (1998-08-13)
list price: US$115.00 -- used & new: US$78.06
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Asin: 0387985425
Average Customer Review: 5.0 out of 5 stars
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Numerical linear algebra is one of the most important subjects in the field of statistical computing. Statistical methods in many areas of application require computations with vectors and matrices. This book describes accurate and efficient computer algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. An understanding of numerical linear algebra requires basic knowledge both of linear algebra and of how numerical data are stored and manipulated in the computer. The book begins with a discussion of the basics of numerical computations, and then describes the relevant properties of matrix inverses, matrix factorizations, matrix and vector norms, and other topics in linear algebra; hence, the book is essentially self- contained. The topics addressed in this book constitute the most important material for an introductory course in statistical computing, and should be covered in every such course. The book includes exercises and can be used as a text for a first course in statistical computing or as supplementary text for various courses that emphasize computations. ... Read more

Customer Reviews (1)

5-0 out of 5 stars Excellent Textbook
This is an excellent textbook for an advanced undergraduate / introductory graduate level course on computational statistics.It is quite accessible to applied statisticians interested in numerical linear algebra, and wouldbe appropriate for an applied statistics course. ... Read more

18. Finite dimensional multilinear algebra. Part I. (Pure and applied mathematics 23)
by Marvin Marcus
 Hardcover: 292 Pages (1973)

Isbn: 0824760778
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19. On the Teaching of Linear Algebra (Mathematics Education Library)
Paperback: 312 Pages (2010-11-02)
list price: US$269.00 -- used & new: US$269.00
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Asin: 9048155495
Average Customer Review: 4.0 out of 5 stars
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This book presents the state-of-the-art research on the teaching and learning of linear algebra in the first year of university, in an international perspective. It provides university teachers in charge of linear algebra courses with a wide range of information from works including theoretical and experimental issues.

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Customer Reviews (1)

4-0 out of 5 stars a meta analysis
Ok, this book is a trifle unusual. There are plenty of books on linear algebra, going back decades. But Dorier gives a meta analysis of the teaching of the subject. Which is something almost universally lacking from other books. Each instance of the latter gives its author's pedagogy. And indeed most texts are fairly adequate at teaching.

Dorier tries to logically discern which methods might be more effective. Kudos for trying. You might not necessarily agree with all that the text suggests. But you get something to ponder, and possibly to improve your teaching skills.

Are there similar books for other branches of maths and science? ... Read more

20. Linear Algebra (Undergraduate Texts in Mathematics)
by Larry Smith
Hardcover: 452 Pages (1998-05-28)
list price: US$74.95 -- used & new: US$47.74
(price subject to change: see help)
Asin: 0387984550
Average Customer Review: 2.0 out of 5 stars
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Deals almost exclusively with real finite dimensional vector spaces & in particular with the principal axis theorem which follows a straight path to its solution.DLC: Algebras, Linear. ... Read more

Customer Reviews (3)

1-0 out of 5 stars Sophomore Level ???
This book is very confusing. The reader is required to make very large leaps to relate example to exercise. Most of these leaps (chap 2-6) involve wading through set theory instead of applying what was covered in a given chapter. It seems as if the questions are designed so the reader must figure out some "trick" in order to solve the problem. This book may be useful if the reader is very familiar with logic notation, set theory, and linear algebra; otherwise, you will probably spend a lot of time learning these things from other sources.

1-0 out of 5 stars A Terrible Textbook
This book is god-awful.The author will use the same symbol to mean many different things, and won't bother to explain the new terminology if he's used it within the past couple of pages.

It's impossible to open the book and pick the important bits out of each chapter - theorems, proofs, and examples all look the same. If you want to find that theorem you think you saw several pages ago, you have to skim very carefully.

The exercises at the end of each chapter are all complete wastes of time. Invert a 5x5 matrix! You won't learn anything useful in the process, but it'll take you a while to get there!

If you want to learn linear algebra, just read mathworld and wikipedia instead of buying this book.

4-0 out of 5 stars A very good on linear algebra!
I found the book to be very thorough and well written. It seemed very easyto understand except for thechapter on Jordan Canonical Forms. However, Iwould not recommend this book to someone who is learning it for the firsttime, unless they have a high degree of mathematical maturity. Try thebeginner book by Howard Anton. ... Read more

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