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Rating: 
Summary: This Book Is Almost Excellent
Review: 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.
Rating: 
Summary: An excellent example of a worthless, useless text.
Review: Instead of wasting money on a useless book written by Serge Lang, I would strongly recommend a Linear Algebra book by Howard Anton and Chris Rorres. Every Serge Lang book I have read lacked examples, or if there were examples, they would be too general and not helpful in learning the material you need to know. His books seem to be more based on theories which to me are not helpful for one to learn calculus. Aside from the way it is written, compared to most other Linear Algebra books, it is too small. His book does not come close to any other Linear Algebra book I have read, so there are obviously many concepts and exercises lacking. When one has trouble in a course, their text is supposed to be helpful and used as a resource, and Lang's books are anything but helpful. The only thing this book will be good for is the fireplace when I am done with it. Unless your a super genius or don't need a text except for the homework problems, I would avoid this book or course which uses is.
Rating: 
Summary: Solid piece of work
Review: Like in other math books by Lang, the theory of Linear Algebra is presented in an axiomatic way, the best way of presenting since The Elements of Euclid. The way in which the theory is presented adds to the beauty. I have read this book as a refresher for Linear Algebra, about 20 years after the completion of a master's degree in an exact science. For me the level was perfect. If you have no experience with Linear Algebra beyond high school, you must first read "Introduction to Linear Algebra" by Lang or some other introductory course. The book under review does not talk about basics like Gauss-elimination. I have seen remarkably few typos. Some cross-references to theorems in other chapters were wrong, though. In all: a very good book and well worth the money.
Rating:
Summary: Serge Lang is a Very Gifted Expositor
Review: Serge Lang is a very gifted expositor. I've read the reviews saying that his books are notorious for their "dryness". At least as concerns this book - that couldn't be less true. 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.
Rating:
Summary: Serge Lang is a Very Gifted Expositor
Review: Serge Lang is a very gifted expositor. I've read the reviews saying that his books are notorious for their "dryness". At least as concerns this book - that couldn't be less true. 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.
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