Linear Algebra

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

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 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 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.