Linear Algebra (Undergraduate Texts in Mathematics)

This book is not only methodical and well written, it is a joy. Every section is a well rounded presentation: Lang clearly and effectively introduces new concepts and patiently develops even the most basic results. But Lang achieves much more: his illuminating examples are stepping stones to a more abstract understanding.

Enjoying this book is like enjoying anything of high quality and craftsmanship. Admittedly, that is not always for everyone to enjoy.


This Book Is Almost Excellent   April 25, 2003
Khalifa Alhazaa (Doha, Qatar)
12 out of 17 found this review helpful

I had this book as the text for my second course in Abstract Algebra, having already taken some elementary Linear Algebra course. I might argue that this is not the best subject for such a course, yet this is very irrelevant here.

All through the class I struggled to understand concepts. I did not. That was not due to the book since I did not even bother opening it. After finishing the course, I realized that the material of this book is of the most importance to anyone planning on continuing his/her grad degree in math, so I decided to read the book. The mission was accomplished in a matter of a couple of weeks.

I do not claim that this is the easiest book to understand the material. In fact, Lang's books are remarked for their dryness. Motivation is almost nonextant. If you, however, have a fairly good background in Linear Algebra (something like the material of Anton's "Elementary Linear Algebra" or the like) and Abstract Algebra (an excellent introduction can be sought in Herstein's "Abstract Algebra") you would much benefit from this book.

The book is a very good book for a second course in linear algebra, that is, it is not a good book for those who had no experience with matrix theory before. The reason is that the book does not mention anything about Gaussian elimination and treats the solutions of n equations in m unknowns using dimension theorems, which is not the standard way of proving existence of such solutions. One more thing is that it does not talk about elementary matrices (one can interpret column or row operations by multiplication of elementary matrices to the right or left). I am not saying the book
is bad, I am saying it is not the right book for a beginner.

The book introduces the basic notions of vector spaces, linear mappings, matrices scalar products, determinants, and eigenvalues and spaces. It then moves to unitary, symmetric, and Hermitian operators and explores their Eigenvalues. Polynomials have a whole chapter followed by triangulation of a linear map. The book concludes with applications of Linear algebra to convex geometry.

I might disagree with the definition of the determinant the author offers, but I would have to admit that his approach is the traditional one.

The subjects of the books must be mastered (or at least absorbed) by anyone who wants to go to analysis (Functional analysis to be precise), Algebra, Geometry, and Differential Equations. To ensure this you should do almost all the exercises of the book since they are so excellent and help a lot in understanding the material presented.


Difficult for undergrads   August 10, 2006
T. Morrison (baltimore)
3 out of 3 found this review helpful

I think the term "undergraduate" is a bit misleading. I think you would have had to have at least one course in linear algebra and abtract algebra to truly appreciate this book. I read it over a summer (as a master's student who lacked any coursework in linear algebra) - kind of as an independent project, and I found it to be very easy to understand. Then again, I had just taken abstract algebra. There were a couple parts that I found challenging though. I love it when he says that the proof or rest of the proof is "trivial" and unnecessary to write. I have heard he does this in many of his books. Overall great book if you have some background.