Customer Reviews

Linear Algebra (3rd Edition)

5 star:		(19)
4 star:		(12)
3 star:		(4)
2 star:		(4)
1 star:		(1)

 

 

LINEAR ALGEBRA, Third Edition, is one of the better books on linear algebra. The material is presented in an abstract and mathematically rigorous fashion. The focus of the book is on the ideas and proofs behind the linear algebra -- its treatment of applications to physics and computer science is almost nonexistent. Topics covered include vector spaces, linear transformations, systems of linear equations, determinants, inner product spaces, eigenvalues and eigenvectors, and canonical forms.

For the aspiring engineering or computer science student, this book is not for you. Basic matrix theory is enough for those fields, and this book is littered with rigorous proofs. There are many other textbooks that present linear algebra from more of an engineering or computer science perspective. As an alternative I recommend that you take a look at "Matrix Analysis and Applied Linear Algebra" by C.D. Meyer.

The style of this book is written in the traditional theorem-proof-example style and is thus geared more towards aspiring mathematicians, especially those who enjoy theory and pure mathematics. Many of the examples demonstrate less than obvious inferences and can be very useful, but of course the meat of the book lies in the comprehensive build-up of linear algebra theory from a mathematically sophisticated point of view.

In summary, a highly recommended purchase for mathematicians. Computer scientists and engineers should look elsewhere though.

This book combines a very rigorous treatment (with a flavoring of abstract algebra) and interesting applications. The presentation is very clear and straightforward. You get theorems, a proof of each one, and curious exercises. Some exercises also challenge you to develop and prove results about some side topics. As you go through the chapters and learn more, you prove further results. Also, this book is the first which presented Jordan forms lucidly and thoroughly. Other texts shove it into the appendix, which is a mistake, since this topic is important.

Finally, the applications are plenty. Standard ones like Markov chains, plus a few fascinating applications, like an entire section devoted to the development of the basics of Special Relativity.

This should be the standard text on linear algebra, instead of that drivel by Strang.

This book has a lucid treatment of the matrix theory and linear trnasformations of differential equations. although the applications are extended upto Hamiltonians, Markov chains and relativity to name a few. All in all an excellent book which prepares the reader for more specific topics according to reader's taste. Worth reading is the 6th chapter for advanced students intending to major in physics or maths. Definitely worth having one in your shelf. Clear presentation and ample examples will encourage your appetite for matrix theory etc.

A few introductory comments are in order: (1) This is *not* intended to be a first look at the subject of linear algebra, at least from the "computational side". (2) This is an undergraduate level text, though typically students will not encounter this material before their junior or senior years. (3) There is some overlap with a graduate level course in linear algebra, though this book is not comprehensive enough for a course at that level.

Ok, now that we've gotten that out of the way...

We used this as the primary textbook as a cross-listed advanced undergraduate/beginning graduate course I took in linear algebra. I had to supplement this book with outside reading/assignments to fulfill the balance of the course requirements. Contrary to what you might expect, you do not need an "introductory linear algebra course" (read that as "linear algebra for engineers") to successfully navigate this book. Actually, much (not all) of the material covered in this book should be discussed in any decent undergraduate course in ordinary differential equations (Boyce & DiPrima's ODE text makes a decent reference).

Here, you'll find that the emphasis is on learning the theoretical side of linear algebra. While there is a chapter (Chapter 3) on basic matrix algebra (wholly unnecessary in my opinion), the main use of matrices here is to express linear operators in a form more suited for computations, e.g., the determination of eigenvalues and eigenvectors. Right away, in Chapter 1, vector spaces are introduced and many familiar (some unfamiliar) examples are given. Just as in an abstract algebra course, you define a list of axioms for vector spaces (later, inner product spaces) and see what you can do with them...quite a lot, as it turns out!

To briefly outline the book: Chp 1: Vector Spaces ; Chp 2: Linear Transformations and Matrices (this is where the matrix is exposed as being a convenient representation of a linear transformation) ; Chp 3: Elementary Matrix Operations and Systems of Linear Equations (some filler content...this should have been left out...better discussed either in an "intro" course or in a numerical linear algebra class) ; Chp 4: Determinants (ok, but should have been condensed into another chapter...come on, we should know how to compute a determinant by now!); Chp 5: Diagonalization (the book really shines here...this is the most lucid treatment of the Cayley-Hamilton theorem I have ever seen); Chp 6: Inner Product Spaces (pretty good, more emphasis on linear operators as opposed to arbitrary linear transformations); and Chp 7: Canonical Forms (the highlight of course being the Jordan Canonical Form).

As I mentioned earlier, you'll learn nothing new from Chapter 3 in particular. In fact, if you had a strong enough intro course in linear algebra, the truly new material is confined to parts of Chapter 5, Chapters 6 and 7. That's partially why I only give this book 4 stars instead of 5. What the book covers, it covers quite well...but it should assume more in the way of prerequisites. Also, Chapter 7, while definitely informative, let me down somewhat. All of the material covered there can be done in greater generality (while still being very comprehensible) in the context of modules over principal ideal domains (basically, think of a vector space over a less specialized ring than a field). It turns out that you lose very little in the transition from vector spaces to modules. Also, believe it or not, it actually clarifies some of the proofs concerning rational forms, since you have much more motivation. This book does do a good job of explaining the differences between the forms: briefly, when you can diagonalize, you get the most for your money...when diagonalization is impossible, try for the Jordan form...when *that's* impossible...you can always fall back on the good old Rational Canonical form (which always exists, regardless of how the characteristic polynomial behaves).

If you look at the reviews for Hoffman and Kunze's linear algebra text, you'll find one by a mathematics professor (sometime in 2007) that is right on the money. Hoffman and Kunze is still the gold standard in theoretical linear algebra. If you're looking for "the meat" this book is missing, you'll find it there. Just be warned that Hoffman and Kunze (at least in older editions) has horrible typesetting, and it definitely takes no prisoners. This book is excellent preparation for Hoffman and Kunze, so it is well worth your time to work through it.

As far as extras goes, there are the standard appendices on material you should already know...sets, functions, fields, complex numbers, etc. Also, there is some interesting material squirrelled away in Chapters 5 and 6. Markov chains are discussed in Section 5.3 (the first look at that topic for me, and quite absorbing). In Chapter 6, expect some material from numerical linear algebra (singular value decomposition, conditioning and the Rayleigh quotient) as well as bilinear and quadratic forms (yawn...but ok) and, much to my surprise, a linear algebra spin on Einstein's Special Theory of Relativity (relax, no physics is required).

To sum up, I do not for a minute regret owning this book...in fact, I wish I had read it sooner. This is the kind of textbook that should be used in the undergraduate survey course to begin with. Leave the computational stuff to the ordinary differential equations course. In fact, being a math teacher myself, I can tell you that elementary matrix algebra is filtering down to the college algebra level now, so anyone who is a math major should already know the basics of row reduction, echelon forms, etc., before they even walk into a linear algebra class!

Ok, getting off the soapbox: buy this book, read it, love it, and remember it fondly when you have to take a graduate course in linear algebra. I found Roman's "Advanced Linear Algebra" a good text to continue with where this one leaves off.

In my experience thus far, I have not come read another introductory algebra that is as comprehensive and thorough as this one. It does not sacrific clarity for the mathematics and similarly it does not sacrifice mathematics for clarity. From the beginning it builds and expects you to keep up as it introduces new topics. There is a definate succession and continuity in this volume which does not exists in many other introductory algebra texts. Furthermore, it presents good proofs and asks for the reader's help where appropriate.

The only aspect of the book that I would critique is its problems. Even though they have somewhat challenging ones, there are none which truly test the depths of ones thinking on the material presented. For example, Spivak does this well in his "Calculus".

Nonetheless, this is a great book. It covers standard topics with a few applications thrown in for good measure. Even so, it is unmistakebly a math book, not a science/engineering text on mathematics. I would recommend this to anyone who want a solid start in linear algebra.

I am currently using this book in an advanced linear algebra course. I am convinced that determinants is the wrong approach to understanding diagonalization and the characteristic polynomial. The traditional approach, which this book uses, is an elaborate time-consuming construction that offers little insight into diagonalization or even into determinants themselves.

If you insist on the determinant approach and you want to rehash every single thing covered in the first linear algebra course, this book is not a bad choice. The proofs are certainly clear, although it is easy to lose the big picture. Again and again, the authors prove that some statement applying to linear maps applies to matrices or vice/versa. Enough already! Often theorems are written in a way that obscures the main point, which is then stated as a corollary.

Important ideas (including definitions) are embedded in the exercises. This is an issue in using the book for self study, although I did not find the exercises to be challenging.

This book blew me away! it's just great, it really explains everything (also an axiomatic approach of determinants which is very important to gain a thorough understanding of what determinants actually mean and which will help you when you are going to study multilinear algebra, exterior algebras in abstract algebra etc), the book gives you a good insight in the stucture of linear operators on a finite-dimensional vector space and provides lots of examples and useful applications (e.g. in economics and physics). There was a comment of one customer which criticized the fact that the theory doesn't offer an explanation of quotient spaces, this is true but in fact this is not important in the area of linear algebra, quotient structures should be studied in abstract algebra, when more algebraic structure has been developed so that one can really understand what quotients of algebraic structures are about. So the book is great, but I would recommend some knowledge about polynomials, fields, algebraic closure, vector space before starting to read this one.

For reference, I have done only a few of the problems and read no other books on Linear Algebra.

That aside, I can still attest that this is a good book. The proofs throughout are short, straightforward, and remarkably free of even trivial errors. The organization is sensible (at least from a theoretical perspective), and any definitions are generally introduced when the motivation has been established. The book is best geared for a math major, but I think the clarity is good enough to make it suitable for physics and engineering majors as well. To keep the book lively there are some well-developed examples in linear diffeq, economics, and einstein's relativity among others. These extra sections can ofcourse be skipped without loss of continuity. As per the problems, they are mostly of a trivial nature (dealing with concrete numbers) with a couple of intermediate proofs towards the end of each section.

My only gripe is that the authors take little initiative to ascribe geometric interpretations to results whenever possible; especially in the chapter on inner products. Frankly, it's easier to remember pictures then verbose thereoms.

If you do plan to read the book, I would recommend two semesters of calculus and possibly a preliminary course in abstract mathematics (sets and proofs).

I used the 3rd Ed. in UC Berkeley's MATH 110: Linear Algebra, then used the 4th while grading homework for the same class the next year. I think the book is fairly comprehensive (though by itself not enough to prepare one for grad school), and very well-written. The exercises at the end of each section span a wide range of difficulty. The book is self-contained, except for a few basic results from the calculus (one has to know the linearity properties of derivatives and definite integrals, i.e. derivative of linear combination is linear combination of derivatives and similarly for integrals), yet does sort of assume prior knowledge of linear algebra. At UC Berkeley students have already taken MATH 54: Linear Algebra & Differential Equations, which includes a brief treatment of vector spaces, linear transformations, eigenvalues, etc. I wouldn't say this book is "not for the faint of heart," as some reviewers put it. I think it's ideally suited--essential, in fact--for entering juniors majoring in the any of the mathematical sciences. If this book is your first exposure to linear algebra, then I highly, HIGHLY recommend chapters 12 and 13 of Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition), and chapters 1-5 of Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications.

One of the best linear algebra expositions, however I would advice some familiarity with matrix algebra and modern algebra before attempting to read this book. The book emphasizes abstract vector spaces, giving special attention to the relationship between linear transformations and matrices. It covers all main subjects in linear algebra from invariant spaces to canonical forms and includes applications to numerical analysis (condition number),statistics(markov chains), and linear differential equations.