Customer Reviews

Linear Algebra (2nd Edition)

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As a professor of mathematics, I was recently assigned a section of our undergraduate linear algebra course; the last time I taught the course was twelve years ago. While doing the obligatory search for a course text, I have been surprised to see how the first course in linear algebra for mathematicians and scientists has "evolved" since I last taught it, at least insofar as that evolution is reflected through available and popular textbooks.

In one of the more popular linear algebra texts currently on the market (I will refrain from naming it), the formal definition of a vector space does not even occur until page 198, and this is not atypical. Looking through half a dozen of the more popular texts, one finds lengthy introductory chapters on vectors in R^n and their properties, basic matrix algebra, systems of linear equations, special algorithms for computing determinants and matrix inverses in efficient time, and significant space devoted to special matrix factorizations, such as the LU factorization. I would like to point out, without passing judgment, that this has not always been the case. Over time, the undergraduate course in linear algebra for mathematicians and scientists has evidently acquired a partial resemblance to the computational, non-proof-based course in "Matrix Algebra" that used to be offered to "casual users" of this area of mathematics at nearly all major universities.

Hoffman and Kunze's book was written for the undergraduate linear algebra course at MIT in the 1960s. Those of us who pursued graduate study in mathematics in the 1970s saw copies of this text, with its vivid purple stripes down the cover, on the shelves of virtually every serious graduate student. Simply put, Hoffman and Kunze was a "standard" undergraduate reference for decades, which continued to inform its readers well into graduate programs or professional careers.

The author of this review did not have the good fortune to use Hoffman and Kunze in a course, but I always had a copy at hand as a reference. My first linear algebra course, taken as a sophomore in the 1970s, used a text by Robert Stoll and Edward Wong (Academic Press, 1968). In Stoll and Wong, the definition of a vector space occurs on page 4, not on page 204. There is no preliminary chapter on basic matrix algebra; these computations are discussed as they arise, in context, when one chooses a basis for a vector space and therefore places coordinates on that space. The entire organization and conceptual structure of Stoll and Wong's book is worlds apart from the texts I have been reviewing of late. The same may be said of Hoffman and Kunze, and indeed of most of the popular linear algebra books from that period of time. This is why I am a bit disturbed when I read reviews that declare Hoffman and Kunze's classic text "outdated," "irrelevant," or "impossible to read." If the younger reviewers are comparing Hoffman and Kunze to most of the popular competitors that have been published in the past five years or so, then they are comparing a remnant apple to a crate of newly harvested oranges.

Against all odds, Hoffman and Kunze remains in print, 46 years after its first apperance. And this in an era when the typical college text remains in print for what seems like less than five years. There is a reason for this longevity. For serious students of mathematics and the mathematical sciences, this text remains invaluable. If one is going to be called upon to actually USE linear algbra in any substantive way (and by substantive I do not mean inverting a matrix or solving a system of two linear equations in two unknowns), then one eventually must learn about such things as dual spaces and double duals, cyclic decompositions and the Jordan canonical form, unitary operators, self-adjointness, the spectral theorem, and multilinearity and tensors. One cannot even find most of these topics in the most popular undergraduate texts currently available on the market; they appear to reach their summit when they discuss eigenvalues and eigenvectors. As a consequence, if a student in an advanced course in, say, differential geometry or differential equations is sent back to his or her linear algebra text to read about dual spaces or the Jordan canonical form, then it will be necessary to abandon the text with which he/she is familiar and refer to a more serious reference like Hoffman and Kunze. How terribly inefficient.

In the spirit of fairness, I must observe that the text Linear Algebra, 4th ed., by Friedberg, Insel and Spence is a currently available undergraduate text that is comparable to Hoffman and Kunze in coverage and rigor. It is an excellent text for a first course for mathematics majors---a true anomaly among a host of weaker competitors. However, the authors may dissuade many would-be users by their declaration in the preface that their text is "especially suited for a second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis." The second undergraduate course in linear algebra is evidently becoming increasingly common; is this because the first course has been weakened to "matrix algebra" and therefore leaves the student unprepared to cope with advanced mathematical courses?

My sincere thanks go out to Prentice-Hall for keeping Hoffman and Kunze in print all these years. Linear algebra is the essential prerequisite for nearly all advanced mathematics, and it is good to see that at least one definitive reference remains available, even as market and societal forces in higher education bring about a clear, demonstrable devolution in the quality of introductory texts on the subject.

This is a fantastic book on linear algebra. Not only does it cover abstract vector spaces through the Jordan canonical form, but takes the reader through complex inner-product spaces and the Spectral Theorem as well. For the reader who has the mathematical sophistication, this is a great book. Excellent preparation for the student planning to attend graduate school.

I got this book for my Linear Algebra class about four years ago. This is a great book if you are getting a degree in mathematics. It won't help if you are just trying to get by the class and don't like math. It is not very practical but if you are looking for a real math book on Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn about Linear Algebra from a rigorous mathematical point of view this is it. This book taught me so much.

This was the textbook they used to use at MIT in the past few decades. Virtually, however, nobody uses this book in a regular undergraduate course anymore. Instead of developing the ideas in the familiar context of the real numbers, Hoffman and Kunze give a more abstract (and general) discussion. For example, the theorems about determinants work in all commutative rings. The rigorousness and the wealth of information are overwhelming for most undergraduates to handle. You will not learn anything if you just glance through the pages. Every line requires deep thought. Down-to-earth applications are not included. So I do not recommend this book for engineers.

I simply loved this book. Hoffman and Kunze have written a very sturdy book that begins with the most basic concepts of linear algebra(such as echelon form) and goes through cannonical forms, inner product spaces, and Bilinear forms. The proofs are complete and at an appropriate level for a first look at the subject. Perhaps one of my favorite aspects of this book was the treatment of dual spaces and tensors. It seems many linear texts deal with one subject or the other but rarely do I see both subjects dealt with in the same book.

The only non-positive comment I would like to make about this book is that its beauty is not in its appearence. When you open the book and flip through the pages you feel a little uneasy. The typeset looks uninviting. Take heart! The beauty of the book lies in its content. Give it a thorough chance, and I don't think you will be disapointed

I highly recomend this book for both learning and reference.

I studied this book between undergraduate school and graduate school in mathematics. It puts linear algebra on a rigorous foundation. I solved hundreds of the problems and found it enjoyable and intellectually satisfying. Before tackling this book, I would suggest the reader get an exposure to matrices and related ideas from something a little more concrete.

A strong grasp of vector spaces is essential for anyone who wants to do mathematics. The study of this book will give you that understanding.

This book is very rigorous and detailed. This book covers what usually is covered in 2 semesters of Linear Algebra, it covers introductory and advanced Linear ALgebra. I would not recommend it for someone who has not ever seen Linear Algebra because of the lack of concrete examples although I would highly recommend it for someone who has had Linear Algebra. This book is all the Linear Algebra you need up to a Master's degree in Mathematics. For Linear Algebra beginners I would recommend the following 2 books because no one book is good enough : "Elementary Linear Algebra" by Robert S. Johnson; and "Linear Algebra 3rd edition" by Fraleigh. Please see also "Linear Algebra, an applied first course 8th edition" page 447 for those of you intersted in Differential Equations.

A comphrensive introduction to the subject of linear algebra that every undergraduate should read. This book explains in detail the concepts that make up linear algebra and will be enjoyed immensely by anyone with an intrest in pure mathematics. Of course, since it is written from a pure point of view, the authors have made almost no mention of practical applications. This is a book soley for the mathematics student. This book has had a wonderful and profound impact on my views of mathematics and I can only give my heartfelt thanks to the authors for writing such a beautiful book. If you have an intrest in linear algebra and operator theory, buy this book! You will have a blast exploring the realm of pure mathematics and thank yourself later.

I have used this book as the textbook for a Linear Algebra discipline. It is considered the "reference" for that discipline in my institute. Although it can be considered the bible on the subject it should not be used alone in a first course of that subject, it is definitely not an introductory book. On the other hand, for a trained student or researcher on the subject it is simply one of the best (if not the best).

I will start with the positives of this book:

This book seems to cover all of the right material that an undergrad must know. This book has complete discussions on everything from matrix manipulation, to eigenvalues, to opertors on inner product spaces, and it even gets into bilinear algebra at the end. Every undergrad wishing to go on to grad school must know just about everything in this book, and the fact that everything is packed into one book certainly is nice. Whenever I want to check how to calculate something, or look up a thoerem, I know for sure that it will be in this book. For this reason, this is a book that will be difficult to outgrow. It is also nice that this book has a lengthy discussion of detemriants that goes well beyond proving that det(AB)=det(A)det(B), sometihng that is difficult to find in many introductory books.

The problem sets are also nice. They strike a nice blend of calculation and theory, and so students will get plenty of practice in both if they attmept all of the problems. The theoretical problems are all pretty good, nothing too diabolically difficult, but there are not too many trivial ones.

And now for the negatives:

One problem that I see with this book is that it focuses a little too much on matrix representation of everything. While matrices certainly do have their uses in computations, it is quite possible for students to learn to rely so much on matrices that they are unable/uncomfortable with the properties that linear transformations have in of themselves. This not only leaves students useless whenever they are confroted with infinite dimensional spaces, but they also use incredibly awkward and difficult proofs, when simple proofs exist that don't rely on matrices. As a quick example, I asked one of my friends to prove that ST and TS have the same eigenvalues for operators S and T. He then proceded to attempt rational canonical form as a way to prove the claim, and when that failed LU factorization. There is a very simple proof that doesn't involve matrices, but I strongly suspect most students would not have guessed that. While this book is not nearly as bad as some others, I definitely see that there is a heavy emphasis here on matrices. To remedy this, I would strongly advise any undergrad to buy Axler's Linear Algebra Done Right in order to see linear algebra from a different (and most valuable!) viewpoint. There is a happy medium between being able to use matrices whenever you need to, and having the foresight to know when better, more enlightening, proofs exist. This book does not cultivate the second.

I think another serious issue, but a rather nebulous one, is that this book is absolutely no fun to go through at all. This may sound a bit childish, but this book is dry, and does not provide anything for it. This will also be most advanced students first exposure to linear algebra. I feel that an author who sets out to write an introductory book has the responsibility to try and "sell" their subject to their students. This can be done by either being relentlessly clear, or by providing multiple/unique viewpoints on the subject. Unlike other difficult/dry intorductory books, i.e. anything by Rudin, this book does not necessarly offer a unique viewpoint or give the reader a higher level of thinking. Just because this book packs in a few extra theorems and topics between the standard undergrad material does not mean that it presents linear algebra well, or in a worthwhile way. A math book should either give the reader a quick education of the basics, or it should provide the reader with a higher level of thinking. This book does neither, and for that reason I really don't think this book is worth the hundreds that amazon is asking.

This book also annoys me for some semi-silly reaons. The typesetting is attrocious! Often times I'll be reading a sentence, only to realize that the letters all of a sudden became slanted, or that half of my sentence is slightly raised. In the modern day of LaTeX, why is this accectable? They often use nonstandard notation for things, like for inner product. And finally, there is the occasionaly typo here and there (even a typo in one of the page headings).

I feel that this book has become the canonical honors undergraduate linear algebra book simply because it was the best there was forty years ago, and it has gained momentum since then. But there have been many more books that have been written, and at least a few of them are strictly better. Also, like I mentioned before, there is a wider variety of books availabe that have unique presentations that are definitely worthwhile. At the time of writing, there are several copies of this book on the amazon marketplace for under twenty dollars. I would advise anyone who is considering buying this book to buy one of those used copies and also a copy of Axler.