Customer Reviews

Introduction to Linear Algebra, Fourth Edition

5 star:		(10)
4 star:		(6)
3 star:		(2)
2 star:		(5)
1 star:		(3)

 

 

I write as a 35-year veteran teacher of mathematics and statistics, at Mount Holyoke College. This semester I am teaching two sections of linear algebra, from Gilbert Strang's Introduction to Linear Algebra, 4th edition. I understand that I'm one of the first, perhaps the very first, to teach from this edition, scooping even the author himself, whose spring semester at MIT began a week after Mount Holyoke's.

In choosing a book for my course, I reviewed more than a dozen choices. In what follows, I'll try to set out why, looking back on the first two-thirds of the semester, I'm firmly convinced that I chose the right book to teach from. But first, here's an excerpt from an e-mail I sent the author a few weeks ago:

I've admired your book ever since the first edition came out, but
in our department we have to wait in line to teach linear algebra,
and this is my first chance to teach from your book. It's hard to
put into words how much I'm enjoying it.

In 35 years, I've nearly always ended up feeling deeply disappointed
with almost any textbook I've tried to teach from. However I've
had the good fortune to find two books I really admire. Yours is
one of those two inspiring books. Thanks to you, I'm having a blast!

Enthusiasm aside, I'll start the substance of my review with four questions, aimed both at students and at teachers. These questions highlight the features I find inspiring - but they are not merely rhetorical: I've tried to formulate questions that should be helpful to anyone trying to decide whether Strang's Introduction to Linear Algebra is the right choice for them. In each instance, although my own answer is a resounding "Yes to choice one!" I can imagine teachers and readers whose preference would go the other way.

* Do you want a book that puts a top priority on the substantive content of linear algebra as a subject in its own right, or a book that uses linear algebra as vehicle for teaching formal proofs?

* Do you want a book whose exercises are imaginative, minimize unnecessary computation, challenge the reader to think about core concepts, and anticipate the content to come, or do you want exercises that closely track the pattern of worked examples du jour, with multiple instances of each type?

* Do you want a book that works hard and thoughtfully to communicate the ideas that unite linear algebra, or do you want a book that has thinned and linearized the content in order to make teaching and learning go more smoothly?

* Do you want a book written by a mathematician with a lifetime experience using linear algebra to understand important, authentic, applied problems, a former president of the Society for Industrial and Applied Mathematics, or do you want a book shaped mainly by the esthetics of pure mathematicians with only a weak, theoretical connection to how linear algebra is used in the natural and social sciences?

To get more technical:

The order of topics in a linear algebra course is often a good indicator of the author's orientation. If linear algebra is trotted out mainly as a show horse, a way to exhibit the sleek beauty of well-groomed mathematics, you won't find the singular value decomposition (SVD) in the index. Dot products, projections, and (horrors!) least squares will come late in the book. Abstract vector spaces, properties of linear transformations, change of basis, and isomorphisms of n-spaces will be prominent.

Alternatively, if linear algebra is recognized and harnessed as a powerful draft horse - and we know from the Budweiser Clydesdales that horses doing real work can compel esthetic admiration - we should expect least squares early, and expect attention to the SVD, as in Strang's book. Strang calls the SVD "a highlight of linear algebra"., and it is. A best-selling competitor doesn't even list the SVD in its index.

Finally, you can tell a thinned version of linear algebra from the real thing by the attention to the duality between algebra and geometry. Intellectually thinner books spend more time on algorithmically-based arithmetic and algebra. These algorithmically oriented books are mere cognitive comb-overs. The more substantial books work systematically to develop in parallel the reader's algebraic skills and geometric intuition.

The contrast stands out in sharpest relief when it comes to eigen-stuff. The deep books, like Strang's, emphasize geometry: does multiplication by A change the direction of x? The comb-overs emphasize Cayley-Hamilton and computation.

In conclusion, I'll borrow from the opening lines of Anna Karenina: Traditional linear algebra books are all alike. Each profound linear algebra book is profound in its own way. At the advanced level, we have a handful of profound linear algebra books, but at this point, at the introductory level, I know of only one: Gilbert Strang's Introduction to Linear Algebra.

George Cobb,
Robert L. Rooke Professor of Mathematics and Statistics,
Mount Holyoke College
Vice President, American Statistical Association, 2005-2007
Lifetime Achievement Award in Statistics Education, 2005
National Research Council,
Committee on Applied and Theoretical Statistics, 1997-2000

I bought this book (the 3rd edition of it) my sophomore year as an undergraduate engineer. I read a couple of sections and then got distracted and didn't pick the book up again until my first year as a graduate student. Before reading this book, my experience with linear algebra had been modest (much to the fault of my undergraduate curriculum), but I soon realized how important linear algebra is to an engineer.

This book was wonderful! I read nearly the entire thing over the course of a month (working a large number of the problems), and since then have referenced it often. The chapter on Eigenvalues, Linear Transformations, and Applications are extremely useful (in the 3rd edition 6,7, and 8). Strang's style is refreshing in the world of dry math books; he really gives you the intuition and excitement behind the math. I find this invaluable as an engineer.

There is a downside to this: the book is wordy for a math book and the key results scattered throughout the text. For this reason I would highly recommend this book for someone without much background in linear algebra, but probably would not recommend it to someone looking for a refresher--a more succinct book would probably be more appropriate. I would also not recommend this book for someone interested in formal mathematics: the book claims informality, and it certainly is informal. That being said, most of the essential proofs are there in spirt, just not set down formally like many other math texts I have used.

I gave the book 5 stars because, although it isn't for everybody (no book can be), it is exactly what it claims to be: an INTRODUCTION to linear algebra, and an excellent one at that.

as an earlier reviewer pointed out, the best part of this book is that it follows along perfectly with Strang's free lecture series [...]
with lecture notes, assignments and of course the video lectures. When I you look at all the "value added" textbooks with CD's full of unhelpful material that takes more time to use than the benefit it gives, compare it to the pairing of this book with the online material, which together make a great learning experience.

I had a very poor background in Linear algebra and I had a really advance optimization course. This book along with MIT opencourseware 18.06 (the course Prof Strang teaches at MIT) saved my semester. OCW is free. I went through some important video lectures and almost the whole book. Very simple english, covers all the basics, tells a bit about Matlab, has a chapter on engineering application examples(didn't go through that part). If you get confused regarding something in the book, which in very few cases you might if your running though it like I did, watch the related lecture and then go back to the book. You'll get the BIG PICTURE... I did. Thank you Prof. Strang.

I used this book in my Linear Algebra class, and my students, even some that failed the course before, told me they finaly understood Linear Algebra.
With all due respect, I don't know how someone can give it less than five stars. Since the beginning, the book develops, in the student, a strong intuition, which allows him/her to develop slowly and steady. Each topic advances the knowledge just the right amount.
Also, the book has a set of companion youtube lessons that are among the best lessons I have ever seen - it is good not only for students but also for professors who want to learn how to became better teachers.

I can only speak for my-self. I am getting a Phd. in physics. Quantum Mechanics is really the application of linear algebra. This book was tailor made to go along with QM. I have heard some people say that there is not enough theory in this book. But this book gives you more than the theory of linear algebra, it leaves you being able to use linear algebra just like you use regular algebra.

After reading this book I learned how to use linear algebra. There were a lot of people struggling with the linear algebra in my Quantum Mechanics class. I want this review to be generic so I will only say that in QM what you observe in the laboratory, is called an observable and it is represented by an Eigenvalue. To understand how to do QM you must have a understanding what is going on mathematically. The math used in QM is linear algebra.

You must be able to use linear algebra just like you use regular algebra. You do not think about the theory behind regular algebra when you use it, you just use it. This book teaches you to use linear algebra in the same way. In Engineering, Physics or any of the sciences you must know linear algebra to be able to advance to higher levels. This book teaches linear algebra so you can use it and in my case it was learning Quantum Mechanics not learning the math behind it.

I do not know the level at which a person that reads this review is at, but once you hit the upper level courses the instructors assume you know how to use (notice I did not say assume you know) linear algebra.

Put simply this book will teach you how to do linear algebra and that says it all. Sooner or later you must put the math behind you so you can learn the subject at hand. Can you imagine having a problem doing regular algebra. You can know all the theory but in the end all that counts is can you do the math and get the answer.

Pros:

1.The coverage of the four fundamental spaces is fantastic. The pictures showing their dimension, orthogonality etc is very very good and easy to remember. And the importance of the understanding of this topic can not be over emphasized. Prof Strang did an excellent job here.
2.The on-line lectures are excellent. Well at least for most of them.

Cons:

The SVD part is not excellent. Prof. Strang agrees that this is the climax of linear algebra. But the presentation of this critical topic is not excellent. The link between the four fundamental spaces and SVD is not so clear. And the related pseudo-inverse is also not so excellent. Unfortunately, the video on SVD also suffers from the same problem. But I would not blame too much on the video. After all, I am not sure if an excellent coverage of SVD can be given within one lecture. But the book should have plenty of space for a good coverage.

For a good SVD coverage, I found the following is good.
1) A good understanding of the four fundamental spaces is very important. Prof. Strang does an excellent job in his book.
2) Some internet articles actually provide good introduction, better than those provided in books
3) An article on-line by Dan Kalman is very comprehensive but not the first thing you should read.
4) Trefethen in his Numerical Linear Algebra has some good (but not complete) description.

I am a fan of Prof. Strang from his online linear algebra videos, so I decided I would try his book. I have found the book very interesting, but somewhat hard to read. The problem, for me, is the typical tendency in math books to ignore intermediate steps. This problem compounds itself in the first chapter on vectors, where a typo in a matrix mars entire discussion, as there is no restatement or step by step deliniation.

Also, notation is nonstandard and inconsistant, and illustrative figures are explained somewhat obscurely. But if your willing to put in the effort, there is a real sense of adventure in the way Prof Strang advances the topics. Also, his likable personal style shines throughout.

This is an excellent textbook for a linear algebra course. Dr. Strang's presentation is given in a very readable converstational style which takes the time to give geometric interpretations and provides proofs for mathematical rigor. The problem sets are well designed to give both practice in the computations as well as guiding you toward some of the more subtle concepts encountered with specific matrices. Being a fourth edition there are almost none of the minor typographical and editing errors that plague first edition textbooks.

Use this book in conjunction with MIT's Open Course Ware web site with a full set of video lectures presented by Dr. Strang which fits perfectly with this text.

[...]

The book is half very good and half with unclear parts. Some topics are explained in a very clear and nice way. Other topics are unclear. One example is in page 334 where the author tries to prove that a symmetric matrix always have eigenvalues signs that match the pivot signs. The authors asks the reader to look at the changing of values while they "moves to zero". I still can't see this "move to zero" in the matrix multiplication. In other sections he talks about a topic that is only explained further in the book. One example is in page 310 exercise 31 when the author tasks you to work with the Cayley-Hamilton Theorem, but the theorem is only stated clearly in a following exercise. But I would like to note that things that are not well explained in the book are better explained in his lectures. So if you choose to buy the book I would recommend you to watch the video lectures, which are nice.