Book Description

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.

Customer Reviews (9)

Pretty bad
This book starts out well, but eventually the chapters are too short and don't go far enough in depth on the more advanced topics.This book isn't ideal for any sort of self-study, as it does not contain enough information to thoroughly educate the reader on many of the subjects without some supplementary instruction.Also does not go into any detail on using Mat Lab or any other form of programming to solve linear algebra, which is somewhat essential nowadays.Wouldn't recommend.

An outstanding introduction to Linear Algebra
This book provides a good companion for an introductory course in Linear Algebra.Mr. Lay's style is very clear, readable, and each concept logically builds on the last.My only concern is that, like another reviewer said there is the occasional gap between the exercises and the examples presented, which may require the assistance of the instructor.4/5 Stars

A very good introduction to linear algebra
The highest quality of a book is the ability to teach yourself from it.Lay's book is very self-teachable because it is written in a non-pretentious, explanatory way, making sure you get the big picture while making sure you can do the little details.It reminds me of Griffiths books in physics.

It is a little proof light, so I can respect that a mathematician who is into analysis might find this book too easy.Problems aren't too hard but aren't too easy for the more conceptual questions.

And I appreciate that the problems are meant to test your ability to understand the material, not do mindless calculations that I know anyone can do.For example, some matrices will just start off already diagonalized for you in later chapters.

This is written from the perspective of a physicist.I thus say to my fellow scientists that this is a great book to gain a good understanding of the linear algebra.If you are an experimentalist who frankly wants to learn only what he needs to in order to get by, THEN THIS BOOK ISN'T FOR YOU.This book develops from scratch everything you need to know for undergraduate physics.Go read a Differential Equations book and learn as you go for the linear algebra.If you're a theorist, this is for you.

Good for the basics
I bought this book in June for a Linear Algebra course I will be starting in the fall. I've been looking through it and reading bits and pieces and from what I see I certainly like this guy's style. I've had completely unusable math books before but this is one of the few that I can use without the assistance of a teacher to at least begin to understand the basic concepts involved. Since I'll be taking the course as a high school student and not asa college student this book seems like it will do just fine to introduce concepts, and, from what I hear, the teacher who is taking over this course in the fall is excellent at explaining stuff so students understand it. Maybe after taking the course I'll change my mind, but for now Linear Algebra and Its Applications is one of the better math books I've ever used.

High Quality Linear Algebra Text!
This book is much better than what I feel some reviewers are giving it credit for. Perhaps they have had better exposure than I have to other materials for comparison, but I cannot say for sure.

What I can say is that this book, for me, nearly took the place of the instructor that I had for the course. The book is literally that good at explaining concepts to you and providing relevant examples. I'm not sure what some reviewers meant when they found a gap between the examples and the problems at the end of the chapters, but I did not have this problem anywhere that I can recall.

What I like most about this book, compared to most of the math books I've seen in my day (more than enough, I assure you) is the format of the writing and the exposition of the problems. There are clear, clear connections made between visual and mechanical aspects of the problem. When you are working a linear algebra problem from the book, you should have a good idea of visually what it looks like that you're doing, where columns and rows are positioned, what elements from each are multiplying by the other and what the end result should look like before you get there. I don't typically get this vivid connection between the abstract visualization of the problem and the actual mechanics in other books, but this one poured it on consistently and I really enjoyed that.

The writing style is also casual enough that you don't feel as if you're being led through chapter after chapter of rote and repetition. Instead, it feels like you're being led further and further toward your understanding of the subject and you really are learning something new along the way at each step.

Regardless of what others say, I think that if you have the financial means and the interest, you should definitely include this book in your repertoire of linear algebra references. It is a gem, a guide and an indispensable reference for the subject. There is no other linear algebra text like it.

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Book Description

This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.

Customer Reviews (23)

Typos
There are several typing errors in the solutions to the solved problems in this book, and some are just wrong. There is a version in which most of these problems were fixed, I am relatively sure that this is not the corrected version.

Linear Algebra
This is the Higher level Linear Algebra book that covers the entire topic range. Helpful.

Excellent summary
I wanted to brush up on Eigenvalues for a compression algorithm and this was the perfect way to efficienctly recall everything from college. Very useful and succinct.

excellent detail
this text is very user friendly. the sample problems are completely worked out, so there is no guessing where the solution comes from. it's a great review book, and very easy to understand for a first time learner of linear algebra. although, i did find a few typos and insignificant errors. for example, when asked to find a parametric representation of the line passing through the point p(4,-2,3,1) in the direction of a vector u=[2,5,-7,8] , the equation for x2 is mistyped as x2=-2+2t. (2t where 5t should be)

Linear Algebra Review
I have found it to be an excellent addition to my library. ... Read more

Book Description

This is a graduate textbook covering an especially broad range of topics. The first part of the book contains a careful but rapid discussion of the basics of linear algebra, including vector spaces, linear transformations, quotient spaces, and isomorphism theorems. The author then proceeds to modules, emphasizing a comparison with vector spaces. A thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory follows, culminating in the finite dimensional spectral theorem for normal operators. The second part of the book is a collection of topics, including metric vector spaces, metric spaces, Hilbert spaces, tensor products, and affine geometry. The last chapter discusses the umbral calculus, an area of modern algebra with important applications.

Customer Reviews (2)

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.

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.
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Product Description
This informally written text provides students with a clear introduction into the subject of linear algebra. Topics covered include matrix multiplication, row reduction, matrix inverse, orthogonality and computation. The self-teaching book is loaded with examples and graphics and provides a wide array of probing problems, accompanying solutions, and a glossary.

Chapter 1: Introduction to Vectors; Chapter 2: Solving Linear Equations; Chapter 3: Vector Spaces and Subspaces; Chapter 4: Orthogonality; Chapter 5: Determinants; Chapter 6: Eigenvalues and Eigenvectors; Chapter 7:Linear Transformations; Chapter 8:Applications; Chapter 9:Numerical Linear Algebra; Chapter 10: Complex Vectors and Matrices; Solutions to Selected Exercises; Final Exam. Matrix Factorizations. Conceptual Questions for Review. Glossary: A Dictionary for Linear Algebra Index Teaching Codes Linear Algebra in a Nutshell. ... Read more

Customer Reviews (29)

Unhelpful, but at least consistantly unhelpful
Gilbert Strang is a wonderful old professor who teaches Linear Algebra at MIT. He is also a terrible math book author. Written, well, like a brilliant man who is dumbing down his ideas for students, the text winds through multiple topics without ever nailing down important specifics. Even the chapter reviews, the 'concise points' of each chapter, are incredibly vague and nonspecific. For example, while discussing LU decompositions, Strang continually speaks of its computational efficiency but never mentions WHY the LU decomposition is computationally advantageous.

The book skimps even a bit too much on theorems, leaving you to believe Strang's hand-waving without walking you through the proofs. I understand this book is about application, but it's a bit too vague just to use for application; I don't need to know how you got to a proof, but I'd like to know why things in a formula works.

Maybe this book is better if you simultaneously take Strang's OpenCourseware class online. However, as a stand alone textbook, it's terrible. I've resorted to using Strang's chapter titles as my online search outline.

And watch the FREE videos on the MIT site
It's hard to rate the textbook by itself, because I'm also watching the videos of the author's class on the MIT Open Courseware website.But no doubt, the two of these together are an unbeatable combination.The website videos are excellent.I do find the book somewhat hard to follow when I try to read it before watching the videos, but afterward, it's quite clear.So maybe I'd rate the book a 3 by itself, but a 5+ in combination with the FREE online videos.Strang is an outstanding lecturer.

What a terrible book!!!!!
I'm not one to be picky about an author's style of writing, but in the case of this book I have to be. It is COMPLETELY FALSE to call this book "introduction to linear algebra", what the title should be is "Gilbert Strang's ideas about linear algebra that no undergraduate in a first course in linear algebra will ever understand". Talk about inconsiderate instead of clearly explaining the steps involved in solving a problem, Strang will simple talk about some example of his personal interest that does nothing to solidify the main idea within the problem. How about the worked examples, he picks some arbitrary matrices and makes it so they can be solved in a single or very few steps that helps a lot when you want to know what's going on within the examples. Homework its freaking great to, the first question within each section is easy and than the second question and beyond is some kind of proof that you have no clue how to even approach because the author assumes you mastered the simple part in one question. I guess when you're a professor at MIT than you're automatically given the right to write terrible textbooks.

The engineer's classic.
People say that mathematical truths never change, and that's true enough. New concepts, applications, and techniques keep emerging, though, so math teaching needs to keep up with the times. Strang has done an outstanding job of keeping this book current and relevant.

It's not a mathematician's math book - this is aimed at people who need results and needs computational techniques more than they need crystalline theorems. That's why it's so helpful to see applications like Markov models, Kirchoff's laws, and Google's analyses of the web. It's also helpful to see examples worked in Mathematica and MATLAB, the tools of choice for desktop exploration of numerical systems. It's startlingly easy to come up with a 100x100 system of equations, and just nuts to try to solve it by hand.

Strang assumes some amount of calculus in this book, something that other books on linear algebra sometimes skip. That raises the bar for the readership, but also opens up topics like change-of-basis in function space, including Fourier analysis. It also allows differential equations to be addressed as linear systems. Even without calculus, though, a reader is exposed to the singular value decompostion, QR and other matrix decompositions, and considerations in performing the computations. I found a few oddities, such as the description of a matrix's condition number. That has great physical meaning when it's taken as the ratio of the matrix's highest and lowest eigenvalues, but Strang gives a definition that I found less intuitive.

Such oddities are rare, though. Even though this book covers many topics, its emphasis is on clear and applicable presentation. I recommend this to anyone studying linear algebra or who, like me, has to brush up on basics not used in many years.

//wiredweird

A wonderful book
As someone who already knows the basics of the subject, I guess I'm looking at things with the benefit of hindsight.However, I needed to shore up my own knowledge of Linear Algebra and thought I might as well turn to Strang for a refresher and a different approach.

The result is that I am truly pleased with this book.His writing is lively and engaging.Linear Algebra has a phenomenal tendency to get dry and Strang does an excellent job of turning the subject this way and that so that one can admire it from every angle.In particular, there are three major approaches in this book that make it stand out.

1.Strang places heavy emphasis on vectors, vector spaces and transformations.This is good preparation for future study in Linear Algebra.This will provide an intuitive understanding of linear operators on vector spaces later.

2.Another reviewer mentioned that the book utilises a discovery-based approach.While this might be a disadvantage when you're in a hurry, the approach prepares one well for learning more theoretically oriented subjects where self-guided discovery is imperative.In this sense, I think the discovery approach is far superior to others and prepares the reader well for future studies.The problems are really fun (although I personally think they are much too easy).Many of the questions require light-weight proofs without undue formalism (not really required at this level).These pseudo-proofs really do help build understanding of the subject.Maths-phobes will not even realise that they're fleshing out the subject themselves.

3.The didactic approach taken in the book is conversational and informal.When added to the freely available video lectures at the OCW site, given by Strang himself, you really have a perfect introduction to the subject of Linear Algebra.The lectures are superb and Strang is an excellent teacher.His enthusiasm and passion for the subject is obvious and infectious.

I really wish I had learned Linear Algebra from this book initially.The book does a good job of encouraging geometric intuition and visualisation.That said, I do not think the book is an ideal book for maths majors.The primary problems being too little exposure to abstraction and problems which are too easy.However, I do believe that the book can be used in conjunction with a more rigorous approach in cases where the latter gets just a touch too dry.There is time to develop the rigour in theoretical Algebra courses at a later stage, with the added benefit that the reader will have learned the experimental approach to learning taken in the book.

I suppose some will find Strang's excitement over Linear Algebra a bit of a pain, but personally I think this conveys the sheer joy of pursuing an intellectual endeavour.I've always bordered on disinterest with Linear Algebra and this has been very much dispelled.I like to be reminded why I chose to study mathematics in the first place sometimes.While I can see that this book isn't for everyone, I really enjoy it and recommend it highly. ... Read more

Book Description

Solved Problem Series

These books help readers review and master what they've learned by showing them how to solve thousands of relevant problems. Perfect for preparing for graduate or professional exams, these detailed reminders of problem-solving techniques show readers the best strategies for answering even the toughest questions, including the types that appear on typical tests.

Customer Reviews (7)

Saved my Posterior
Almost a decade ago, I ill-advisedly decided to tackle a course in Linear Algebra via a correspondence course offered through a prominent Canadian university. The textbook that accompanied the coursewas so advanced that the pages might just have easily been left blank for all the good they were doing.Having spent a tidy sum on the course, and needing the credit for another course in the next term, I was at my wit's end.I had used the "3000 Solved Problems in Calculus" to generate exceedingly high marks (A) in Calc I and II, so I was already familiar with this brilliant series of learning aids.True to form, when my "3000 Solved Problems in Linear Algebra" arrived at my doorstep, its effects were immediate.If you haven't any experience with proofs, this book will gently lead you by the hand through most proofs required by your standard Linear Algebra course.Once you see the process in action, its very easy to go it on your own after a bit of practice.If you had a great deal of time to go it alone, Linear Algebra would be a pleasant little intellectual diversion.However, in the university setting, when the pressure is on, who has the time to think this matter through to a successful conclusion?This is where the book shines.It teaches you the technique by examples, and gives you the confidence to really excel at the subject in a very short period of time.As well, I highly recommend Anton's Linear Algebra as a primary learning resource.

Linear Algebra Review
It is good to have this book as a reference in my library.

Very good book, worth the price
I got an A in my Linear Algebra class.Part of it may be due to the countless hours upon hours that I studied every week.However, the tools at my disposal helped make the quality of the time worthy.This book is another valueable tool to keep in your toolchest and I'd recommend this as an accompanyment to your class.They do a great job with Vector spaces and Determinants as well as Gaussian elimination.I particularly liked some of the "trick" questions that they stuck in from chapter to chapter (great study aid as many teachers like to stick these types of trick questions).My only complaint is that they prefer to calculate the basis of a row space by non-transposition (whereas our teacher (in our course) used the transposition technique) so I could not really go by that section (thus the **** rather than ***** review) but I can't really fault the author as teachers have their preferences as to how they want their students to calculate certain answers.

I recommend this book.Heck, it's only $24 (you can get a used copy for half the price) and I suspect you'll spend "more" time with this book than you will your actual $100+ textbook (I know that I did).

I'd also like to recommend that you hunt down a used copy (or go to your library and see if they have it) of "Elementary Linear Algebra" by Howard Anton.We had to use the Grossman book (along with its confusing examples and below-average way of showing proofs).After getting frustrated with the Grossman book (and only using it to do the homework problems (but not using the book to actually understand the concepts as the book was way-below-par in doing so) as our teacher assigned us numerous problems), I stuck mainly with the Anton book (any edition after (or including) the 4th edition will do) and this book to get me thru the course with an A.

Get this book though.Even if it helps you understand just one topic more clearly then your textbook (and I'm sure it will help you with dozens), it will have paid for itself.You can never practice enough (as Linear algebra will take alot of your time) and this book will come in handy with their explanation and sample problems (that so many textbooks neglect to show).One word of caution though is that the author prefers to solve some of the problems in ways that your teacher may not.Never-the-less, a good book.

Good book but without index.
This book is a good complement for any course of Linear Algebra.
Is very clear and complete. Unfortunately it does not have a detailed index at the end of the book, in spite of being announced in the front cover. Only a Table of Contents is available. Other Schaum's Solved Problems books have a really great index that help students locate the type of problem they need to solve.

Jorge O. Henao

Outstanding book
This book contains 3000 Solved Problems (duh) for those who learn best by example.If you need to catch up on your linear algebra knowledge, or currently taking a class, this book will take the abstract concepts and lay them out in example.Some of the problems do ask for proofs (1 out of 15 or so) of certain theories.But don't let that scare you, if you don't like proofs.Practicing answering these proofs with the answer right in front of you, will not only help you construct proofs in the future, but also will also help you organize the thought processes and give you a better concept and intuition of the material.
The range of material covered is more than you will find in the lower undergraduate class.If you take the time to work through this book, you will be a master of the topic with a much wider foundation and with many different approaches to same problem.If you are a veteran of Linear Algebra, this will refresh your memory and may add new elements to the subject that you weren't presented initially, such as manipulations of a matrix in C not just R.It will also help you catch up on the little "details" which you might have absorbed for the duration that you thought you would be tested on it, but after such time the information vanished into the void of forgotten math. ... Read more

Customer Reviews (43)

Incredibly Accessible
I used this book to teach myself linear algebra using a lesson plan developed by a teacher. A few things I found really helpful:

Lay constantly admonishes the reader to carefully study the text, by reading and re-reading. He understands that students learning linear algebra are likely to to be embarking on more abstract/advanced mathematics in the future--and he cares enough to teach good study skills as well as math. I've followed his advice, and found that with proper effort, I am able to teach myself.

Many of the questions are conceptual or True/False, which helps me to retain new concepts.

The study guide (sold separately, but very important if you're self-teaching) only includes answers to odd-numbered exercises. However, the answers are nicely detailed.

I especially appreciate that Lay refuses to flat-out offer answers to conceptual and True/False questions. Instead, there are hints and references to page numbers where helpful information can be found. This is important, not only because it (again) helps reinforce good study skills, but it encourages students to really try a problem before giving up and looking up the answer.

The introduction of new concepts is always followed by one or more examples, which helps to link theory to practice. In general, there are many examples that illustrate good problem-solving techniques. Proofs are detailed and well-justified, and there are some simple proofs that are left as problems in the book. These proofs are simple enough to not be terribly daunting to the student who is new to more abstract mathematical ideas.

For the student who would ask, "well what is it good for?" there are plenty of sections dedicated purely to applications of material learned in previous sections.

Detail-oriented methodical learners like myself should benefit from this book.

a good teacher helps...
covers 75% of the detail and leaves the 25% up to you. it lacks clear explition is vector space secotion (row col vector nul basis etc). it seems like you do the first coumple of sections not understanding what you are getting at but when you get to chapter 6 or so the past stuff finaly makes sense. I found the book hard to understand when it was talking about maping, one to one on too. i suggest a TI-83 or higher and the solutions manual. The book also lacks pictures showing dimention row col nul basis etc.

Choose Another Book
The organization and the treatment given to most subjects are well below what one would expect for a Wikipedia entry on each respective topic.

Anyone who uses this book for any kind of self-study is unlikely to get more out of it more than a few examples on multiplying matrices, row reduction, and some poetry about eigen-spaces and vector spaces.

Take a look at ``Linear Algebra Done Right'' by Alxr.

meh.....
This was the textbook that my school has been using for its first and second year Linear Algebra courses. I found it somewhat terse. It seems like it could be a great book for someone already familiar with linear algebra or the logic aspect of mathematics; I don't recommend it for anyone who is not very familiar with proofs. Fortunately, it doesn't lack examples of how to solve computation problems and solutions to the odd-numbered problems are provided in the back of the book. Unfortunately, when it comes to the proof problems, instead of just providing the reader with the answer like most other texts, the author will either mention a hint or refer the reader to the study guide (mine came with a disk which has to be installed on a computer), which could be a problem for readers not familiar with how to do proofs. Serving as a somewhat of a mediator between these two extremes are true/false questions which accompany each chapter in each section of the text. While those start out somewhat fun in the beginning, they can also get tiresome and repetive very quickly. The solutions to these, like the proofs, are not provided, but fortunately, all one has to do is read the text carefully in order to determine the appropriate response.

Shameful Plugs for Study Guide
Perhaps the most important questions at the end of each section are the true/false questions dealing with the concepts developed in the section. The computational problems are good for learning the algorithms but the true/false questions represent what you need to know.

After working through them and justifying your answers you turn to the back of the book and look for the correct solution but instead you find something like "Check your answers before referring to the study guide." Unfortunately the study guide is sold seperately so many people, including myself, choose not to buy it, nor should we have to. It makes the author seem like he cares more about selling book then teaching students.

I took Linear Algebra as an online course and there it is even more important that the material in the text is learned from the reading, as there is no physical lecture to supplement it. I would be more understanding if the author included just the answers in the text and left the justifications for the study guide, but to plug the study guide instead is just plain irresponsible. ... Read more

Book Description

Customer Reviews (11)

Not the best, but, then again, nothing is.
I couldn't really find a truly good, comprehensive linear algebra textbook.This is one of the better ones.You don't even want to touch one of the worse ones.

Outstanding Book
I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them.However, there is no book I have found that has such a deep and clear exposition of determinants.The first chapter alone makes this book worth buying.

However, there's an incredible amount of material in this book, and the later chapters are just as valuable.This is a dense book, but it is fairly easy to read once you get used to the style.I would recommend it to anyone learning linear algebra for the first time, as well as to people who want a deeper understanding or a different perspective.

Like I said before, this book is particularly useful when combined with a complementary text such as Axler, which provides a completely different approach to the subject.This book may come across as a bit old-fashioned, and some might say the material is obsolete, but I believe that everything contained in the book is useful, if only to give the reader a deeper understanding of the why's and how's of linear algebra.And plus: you can't complain about the price!

Great book
This book has several good points.First, it is extremely affordable. Second, it covers all the typical topics in a typical undergraduate linear algebra course within the first 4 chapters or so. This makes it a great reference. I bought it as a supplement to my old linear algebra textbook, and it is great for that purpose.In addition, it continues on to more advanced topics which may not be covered at the undergraduate level; for example, the heavy emphasis on determinants and the more rigorous treatment of spaces, leading to affine spaces. I also have Shilov's book on real analysis, so I like the concise yet thorough manner of the author.

One of the best books
I am a graduate student and wanted to learn linear algebra for fun. This book provided me with one of the best coverage of the subject. It is supposed to be meant for upper level undergraduate courses or first year graduate studies .It is meant primarily for mathmatics graduates it starts right from basics and the proofs given are most succint and to the point. People who want to intutively learn the subject I would refer then to the book linear algebra and its applications.

all and all one of the best texts on the subject and compared to other books it gives more than other books with high price tags

Nitish

Excellent book
This is very complete and clear book . Excellent for applications. Treats topics usually omited in most linear algebra books . Could be used as reference . ... Read more

Book Description
Taught at junior level math courses at every university, Linear Algebra is essential for students in almost every technical and analytic discipline. ... Read more

Customer Reviews (15)

A great book for students
This book is really clear. It's a lot easier to understand than the book we are using for the class and has helped me grasp the lectures better.

Too many typos in the examples
There are so many typos in the examples that I find the book useless. For example, on page 13, the number 13 in the matrix suddenly becomes a 19, and remains the same throughout most of the example. On the top of page 25, the matrix multiplication has three typos! I cannot recommend this book to anyone. The author should have taken more time to check his work.

Not for a real college level course
I bought this book after reading tons of positive reviews praising it. It was meant for a preview to my Linear Algebra and Applications class and as an alternative reference. I read through about half of the book before the semester began, but we pretty much learned everything I read from this within two weeks. I gave up on it and it was just sitting there until I decided to further clarify to myself the concept of kernel and image, and all the proof questions associated with them. And guess what, this book doesn't even address these concepts. Now I have no reason to even open this book because real college level courses go well beyond this depth and relying on the information in here for tests is just an academic suicide...

Not a Self Teaching Guide for first timers!
The book spills out formulas without adequate explanation. I don't learn unless I understand. It's a good book if you want to review concepts and formulas learnt elsewhere! The examples are simple and to the point, which is good.

Overall, don't buy the book if you want to learn Linear Algebra for the first time, but get it if you want a refresher.

Okay, but not as good as the "Dummy Books" collection

Exactly as stated in the title... should have purchased one of the "Dummy" books. ... Read more

Book Description

Renowned for thoroughness and accessibility, this book offers a challenging and enjoyable study of linear algebra that is infused with an abundance of applications. Balancing coverage of mathematical theory and applied topics, concepts are explained with precision so that all readers can understand the material. Worked examples are heavily integrated into each chapter. The book stresses the important role geometry and visualization play in understanding the subject.

Customer Reviews (17)

Worst math textbook.
Had to use this textbook for an intro-level linear algebra class. Totally incomprehensible. Strange math jargons came out of nowhere at the beginning of the chapter without a proper introduction. Lack proofs and explanation how certain methods of computation were made. Emphasizes on applications that don't make sense. Unless you're an engineer, I would not recommend this textbook to beginners who are just starting out linear algebra.

Little or No Examples
I found this book very difficult to follow along without a teacher to explain. Little or no examples. Book is full of proofs. This would be a perfect book for a person to brush up on there skills and review the material.

Not the best but not the worst either
Background:I am an Aerospace Engineering undergraduate student with a strong foundation in math.

Review: To be honest, I was looking forward to giving this book 1 star when I was done with it.However, after reviewing what I learned, I found this book isn't as bad as I first thought.First of all, this is not a good book for beginners or self-teachers.It skips over too many steps when discussing some topics.Second, the textbook is written in a very dry and dull manner.This is true of many math books, of course.While this doesn't exactly make a math textbook useless, it works against keeping the reader's interest.Finally, it's layout isn't exactly friendly.Important details tend to blend in with the rest of the text.In my opinion, the important points should stand out.
So, with those points having been made, I feel Leon's Linear Algebra book is more suited for an intermediate class.It definitely wasn't worth the money (what textbooks are?). If you'd like a good beginner or self-teaching book, check out Poole's book instead.

good introduction
A very good introduction to linear algebra. Explanations are very clear. A very good exposition on vector spaces. Overall a good buy for starting in this most interesting subject.

Good Book for the Most Part
Overall, I thought was well laid out.I would probably like to see more practical application of linear algebra to real world problems. ... Read more

Book Description
David Poole's innovative book emphasizes vectors and geometric intuition from the start and better prepares students to make the transition from the computational aspects of the course to the theoretical. Poole covers vectors and vector geometry first to enable students to visualize the mathematics while they are doing matrix operations. With a concrete understanding of vector geometry, students are able to visualize and understand the meaning of the calculations that they will encounter. By seeing the mathematics and understanding the underlying geometry, students develop mathematical maturity and can think abstractly when they reach vector spaces. Throughout the text, Poole's direct conversational writing style connects with students, and an abundant selection of applications from a broad range of disciplines clearly demonstrates the relevance of linear algebra. ... Read more

Customer Reviews (12)

bad service
The book was in excellent condition but it took forever to receive the book and I did not have it when I was supposed to.

Excellent book for beginners in linear algebra
I'm going to make this simple: This book saved me!

I'we been studying linear algebra past semister, and the book the teacher assigned to us really wasn't explaining things well enough.I got this book one week before the final exam, and I can surely say that it is the only reason why I passed this class!

The book is very well written, and the author always seems to be ahead of your head, in knowing what parts might get you confused or what questions you might have in mind.It seems like every time I stumbled on something, the next sentence exactly explained what I was wondering about.

Buy this book, you will not regret it!

Excellent text for self study or coursework
I am a control systems engineer and deal with the design of physical systems, both linear as well as non-linear. I constantly use this text to refresh material related to linear systems and find it an indispensable resource. Here's why.

The text is very clearly written and the exercises are excellent. Dr Poole does not obfuscate core issues with lengthy proofs. Which is not to say proofs are not presented - they are, but in a very accessible (read "geometric") form. I would strongly urge serious students of the subject to work through the problem sets - they will bring a deeper level of understanding.

Highly recommended to students as well as professionals.

A lifesaver
As a computer science graduate student without the proper math background, I've been cramming mathematical concepts into my brain for the last few years.And this book, every time, has provided the most lucid, digestable and yet usefully rigorous explanation I have found for whatever concept I'm learning.The four-page explanation of Markov chains, pp. 217-220, e.g., seems very hard to top.The thing is, in CS especially perhaps, you learn to use math as a tool when you have that "ah-hah" moment of intuition.When you get the idea of what *the point* of the particular mathematical tool is.Then you can apply it and use it to think about other problems you might want to solve.Math as a way of thinking.Which is where the clarity of writing and explanation in this book comes through for you.Which is not to say the problems aren't excellent as well, because they are, and you have plenty of opportunity to practice the manipulations associated with a given concept as well.I can't recommend this book enough.

The best Linear Algebra book. ever!
This is a must have for math lovers. the book has colourful graphs and interesting equations and in addition it has interesting problems for the whole family to solve... If you're here to buy the book for your course, it has the answers in the back for those who don't understand math. Nice book to look at when you're in the Linear algebra class. Pure fun! ... Read more

Book Description

The hallmark of this text has been the authors' clear, careful, and concise presentation of linear algebra so that students can fully understand how the mathematics works. The text balances theory with examples, applications, and geometric intuition.

• New! Learning Tools CD-ROM will be automatically packaged free with every new text purchased from Houghton Mifflin.
• New! Section 3.4, now named "Introduction to Eigenvalues," has been broken into two separate sections to provide more emphasis on the early introduction of eigenvalues. The new Section 3.5, "Applications of Determinants," covers the Adjoint of a Matrix; Cramer's Rule; and the Area, Volume, and Equations of Lines and Planes.
• New! All real data in exercises and examples have been updated to reflect current statistics and information.
• New! This edition features more Writing Exercises to reinforce critical-thinking skills and additional multi-part True/False Questions in the end-of-section and chapter review exercise sets to encourage students to think about mathematics from different perspectives.
• New! Additional exercises involving larger matrices have been added to the exercise sets where appropriate. These exercises will be linked to the data sets found on the web site and the Learning Tools CD-ROM.
• Guided Proofs help students successfully complete theoretical proofs by leading them, step-by-step, through the logical sequence of statements necessary to reach the correct conclusion.
• Text-specific web site features student and instructor resources, including text Chapters 8 through 10; MATLAB exercises for Chapters 1 through 7; data sets for new matrix problems; and a Graphing Calculator Keystroke Guide featuring the TI-83/TI-83 Plus, TI-86, TI-89, and TI-92 models.

Book Description
This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation.

• Clear, accessible, step-by-step explanations make the material crystal clear.
• The authors spotlight the relationships between concepts to give a unified and complete picture.
• Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues.

Customer Reviews (4)

A decent textbook
The book isn't the best organized in some regards, but it's decently written and decently eyeball-scan-friendly.

The ninth edition isn't as good as previous editions
I've been using Anton for my linear algebra classes since 1984, and have gone through nine editions.The ninth edition seems much more confusing than previous editions.He often moves text around without seeing that it makes sense in the new context: a theorem will use notation that isn't introduced until later, for example.He comes back to eigenvalues and eigenvectors over and over, looking at them from different perspectives.This should work if done well, but instead it confuses my students.I'm going to look for another book for the next time I teach the class.

No practical application....
This book mostly consists of a bunch of long droning proofs and severely lacks any real world applications.If your looking for an enjoyable learning experience, this is not the book.

clear exposition of ideas
Anton gives a skilled exposition of a first course in linear algebra. The concept of a vector space is developed. From which the ideas of vectors and matrices naturally flow, where the latter map from one vector space to another. There is a nice stress on the student grasping a geometric picture. This helps her develop a qualitative understanding of the concepts.

Standard results, like the rank-nullity theorem are proven. And the utility of determinants is shown. Later in the book, calculus applications are discussed. The book assumes you already know some simple calculus. Then, eigenvalues and eigenvectors are derived for linear differential systems. Very important in engineering and physics. ... Read more

Book Description
This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation.

• The applications version features a wide variety of interesting, contemporary applications.
• Clear, accessible, step-by-step explanations make the material crystal clear.
• Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues.

Customer Reviews (1)

disapointed
It contains solutions only for selected odd exercises, and most of the solutions are very abbreviated.Despite these setbacks it is still better than no solutions manual. ... Read more

Book Description

This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms. For statisticians and engineers.

Customer Reviews (28)

Good alternative to Hoffman and Kunze
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.

Very good presentation and many exercises
Overall this was a very good book on Linear Algebra. The presentation of proofs were very good. Also there were many exercises at the end of each section. My only complaint was that there weren't as many answers in the back of the book as I would have liked :)

wordy, tedious, and unhelpful book
I have read this book, cover to cover, three times and each time I hate it more. It is a horrible book to learn linear algebra from. It uses 5 pages, to state what could be said in 2 paragraphs, forcing the reader to get frustrated trying to understand a simple concept. Many of it's proofs are unclear, needlessly confusing, and provide no intuition. I hate it. It has this afual concept of using a "dot diagram" to explain Jordan Conical form, really a very simple concept, that is explained horrifically. It's a very difficult read, not for new commers to LA, and it focuses a lot on the tedious computational side of linear algebra. Really, must we say "linear Transformation defined on a finite dimensional vector space over a field F" over and over again? Isn't there some sort of appreviation? I recomend Sheldon Axler's book, he does LA right, and the proofs provide useful intuititon, and he doens't waste time when explaining things. This book only made sense to me after I read Axler's book.

Excellent Book
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.

An excellent book, but not for everybody
In the great game of textbook-writing, there is always the question of which textbook is "definitive", "comprehensive", yet also "accessible" and a "pleasant read". I will say that no textbook can possible exist that can cater to everybody. Such is the case with Friedberg's "Linear Algebra".

Rather than praising or complaining about the book, I will instead describe exactly how the book is written, and the reader can determine for himself/herself whether this is the book to buy.

The book is written "bottom-up". Instead of introducing a general concept, delving into detail, and then expanding again, in a sort of holistic approach, the book starts very rigorously. He begins with definitions and axioms. This includes definitions for the numbers "0" and "1" and their axiomatic properties.

Using these axioms and definitions, he proves basic theorems. He will then demonstrate several examples that relate to these theorems. In the exercises, he will ask you to prove corollaries to these theorems, or perhaps little theorems, that may help in later proofs. He will also introduce definitions and completely new types of problems in the exercises. Thus, the exercises are NOT supplementary - they are essential to the reading.

Subsequent chapters will repeat the process. He will introduce new definitions, possibly utilizing new knowledge from theorems proven in previous chapters. Then he will introduce and prove several lemmas and theorems. He will introduce a few examples for these theorems. Then exercises. Etc.

Most importantly, the author expects that the student learns through proofs. He expects that, by the very fact that you are able to prove something, you understand the material on a deeper, analytical level. If this is how you operate, this book is for you. If you can't understand a topic unless you see how it can be proven directly from what you already know, then this book is for you. However, it is rare to see concrete examples that are applications. Rather, the author presents everything in an all-encompassing, general manner, and leaves it to you sort out the details and specifics. This can be potentially unhelpful.

Overall, if the proof-based approach is what you're looking for, I haven't seen any book that presents the topic of Linear Algebra in such a concise fashion. Now I leave it to the reader to decide whether this is the book to buy. ... Read more

Book Description
Designed for first-year graduate students from a variety of engineering and scientific disciplines, this comprehensive textbook covers the solution of linear systems, least squares problems, eigenvalue problems, and the singular value decomposition. The author, who helped design the widely used LAPACK and ScaLAPACK linear algebra libraries, draws on this experience to present state-of-the-art techniques for these problems, including recommending which algorithms to use in various practical situations. Algorithms are derived in a mathematically illuminating way, including condition numbers and error bounds. Direct and iterative algorithms, suitable for dense and sparse matrices, are discussed. Algorithm design for modern computer architectures, where moving data is often more expensive than arithmetic operations, is discussed in detail, using LAPACK as an illustration. There are many numerical examples throughout the text and in the problems at the ends of chapters, most of which are written in MATLAB and are freely available on the Web. ... Read more

Customer Reviews (14)

Great Book for Self-Learning
I am a second year PhD student in Operations Research and for long I had been looking for a book in linear algebra to help me learn it myself (as I see that I need it no matter what research I want to do. It's just a good tool to know). One of my friends recommended this book to me, I got it and I am very happy with it. The book is great in different ways:
-it is in the form of short lectures and for me who wants to learn linear algebra step by step, this is a perfect approach. You will have a 5-6 page lecture so whenever you start, you are set to finish that lecture.
-It gives you intuition and understanding about what is really happenning geometrically which is amazing. To me, it is very important to have the "feeling" of what is happening because it is only then that you can think about bringing your real problem in this framework.
-The examples in lectures clarify the subject while exercises give you a chance to learn even more.
If you are new to linear algebra or know it but want to refresh your mind on intuitions and systematic thinking, I highly recommend this book.

Must be strong in Linear Algebra to use this!
The reader msut have a strong grasp on linear algebra before using this book.Many algorithms are written in pseudo-code which is nice, but sometimes important details lack.I used this book as a required text in a graduate level course.

Fantastic book, with great insight
I can't speak to the entire book, as I've only made significant use of the section of matrix solvers. Having said that, his explanation of Krylov methods was the most clear and well organized I've ever seen. His book is the first I've seen that so nicely ties together all such methods. It's true that his book is probably not going to be enough if you are planning to focus on this as your research topic. But for those of us who simply need to apply the field to their research, it is the best book I've found, and he goes out of his way to be helpful to the practitioner, a rare thing in a math book. (For example, he has a wonderful flowchart in Chapter 6 providing a rough guideline for selecting a linear system solver based on the properties of one's problem.)

great math text
I used this book at NYU in a graduate class on numerical linear algebra and it was great. The book is incredibly clear, starts from the basics and just goes from there. You won't be lost or feel like it has too little (and I usually have one of those two feelings about a math textbook).

The book is focused around matrix decompositions and does quite a bit of theoretical matrix algebra before it gets into accurate computation of decompositions, what this means and how various algorithms achieve it.

The theorems are clear and the proofs concise and easy to read.

Numerical Lineal Algebra not much on applications
This text may be OK for math theory types, but for engineers wanting to know applications and how to use matrix algebra it was extremely lacking.There are very few examples, only proofs.Hardly any probelms with actual numbers are solved.I only bought this book for a course I was taking and I ended up hardly using the book at all because it was just too difficult to interpret.Not recommended for those looking for applications. ... Read more

Book Description
Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than deduces. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics. ... Read more

Customer Reviews (68)

Good to buy only if your instructor follows it or if your have firmly determined to master linear algebra.
My first complaint about this book is it is too expens