Rating: 
Summary: FRUSTRATING
Review: First of all, a little background on myself. I am a college sophomore and an overachiever in mathematics for my entire life. I am currently taking a Linear Algebra course that uses this book. We have just finished chapter 2 and I am ready to throw my book in the fire.
Before I talk about why I hate this book I will first give some positives.
1. The book is very concise. A lot of material is covered in relatively few pages. There are not many pictures or graphs to clutter up the pages.
2. This book covers the material very thoroughly. Everything is defined and everything has a clear proof, except for a few theorems which are left as exercises.
Now for the bad...
1. The thing that drives me insane about this book is the lack of decent examples. Since this is new material to myself and many others, it really helps to aid understanding if there are well-explained examples to accompany the new information. Most of the examples in the text are either WAY too simple, or are left thoroughly unexplained. Sometimes I feel like a toddler trying to learn addition, but all I have to go on is that 0+1=1.
2. There is no answer key. If there were an answer key for the problems it would more than make up for the lack of examples, but instead we are left with about 5% of the answers to the simplest exercises.
3. There is little explanation of the computational aspects of linear algebra. This goes hand-in-hand with the poor examples. If the text spent as much time explaining how to use the material as it does explaining where it is derived from then it would be a lot easier to understand.
That's all I can think of for now. This is definitely not the worst textbook in the world, but my opinion is that the bad outweighs the good and it should be avoided by anyone new to Linear Algebra.
Hope this helps.
Rating: 
Summary: Excellent Text for Theory
Review: For reference, I have done only a few problems, and haven't really read other books on Linear Algebra.
That aside, I can still attest that this is a superb book. The proofs throughout are clear, short, straightforward, and remarkably free of even trivial errors. Definitions are generally introduced when the motivation is sufficient and the organization is undoubtly sensible (from theoretical perspective). Despite the heavy emphasis on theory, there are well-developed examples as diverse as linear diffeq, economics, and einstein's relativity. These extra sections can be skipped without loss of continuity. As far as the problems go, they progress from trivial to more difficult, interesting proofs.
My only gripe is that the authors take little initiative to give a geometric interpretation of results. especially in the chapter on inner products. It is much easier to remember visual pictures in your mind then a wordy thereom.
If you plan to read the book, I would recommend two semesters of calculus and a preliminary course in abstract mathematics (sets and proofs).
Rating: 
Summary: Does rigor need to be so pedantic?
Review: I am currently using this book in an advanced linear algebra course. I am convinced that determinants is the wrong approach to understanding diagonalization and the characteristic polynomial. The traditional approach, which this book uses, is an elaborate time-consuming construction that offers little insight into diagonalization or even into determinants themselves.If you insist on the determinant approach and you want to rehash every single thing covered in the first linear algebra course, this book is not a bad choice. The proofs are certainly clear, although it is easy to lose the big picture. Again and again, the authors prove that some statement applying to linear maps applies to matrices or vice/versa. Enough already! Often theorems are written in a way that obscures the main point, which is then stated as a corollary. Important ideas (including definitions) are embedded in the exercises. This is an issue in using the book for self study, although I did not find the exercises to be challenging.
Rating:
Summary: Does rigor need to be so pedantic?
Review: I am currently using this book in an advanced linear algebra course. I am convinced that determinants is the wrong approach to understanding diagonalization and the characteristic polynomial. The traditional approach, which this book uses, is an elaborate time-consuming construction that offers little insight into diagonalization or even into determinants themselves. If you insist on the determinant approach and you want to rehash every single thing covered in the first linear algebra course, this book is not a bad choice. The proofs are certainly clear, although it is easy to lose the big picture. Again and again, the authors prove that some statement applying to linear maps applies to matrices or vice/versa. Enough already! Often theorems are written in a way that obscures the main point, which is then stated as a corollary. Important ideas (including definitions) are embedded in the exercises. This is an issue in using the book for self study, although I did not find the exercises to be challenging.
Rating:
Summary: A great text for introductory linear algebra.
Review: I had trouble with the first Linear Algebra class I took, which used a different text. I had a very incomplete understanding of the core elements of Linear Algebra, and this text corrected all of my misunderstandings with its examples and language. I liked the extensions, like Leontief models, that gave applications for the material. I also liked the integration of Linear Algebra with Differential Equations. Even though I had this class in Fall of 1997 (taught by Dr. Friedberg at Illinois State University), I still own this book and keep it on my shelf.
Rating:
Summary: An Extremely Frustrating Book
Review: I have never seen a book which hides so many important details (and even basic definitions) in exercises. This book was terrible to learn from and even worse as a reference. It also omits or glosses over a number of important topics including quadratic forms and matrix norms.
Rating: 
Summary: Elementary Linear Algebra
Review: I used this book in a junior-level course about a year ago. While the book is well written and the proofs are lucid, the exercises lack in providing any sort of challeng. Most of the proofs that students are asked to write in the exercises are very elementary, and follow directly from the definitions, theorems, etc.
As for the authors' habit of incorporating definitions within the exercises, it makes me wonder how important they really are. It seems to me that if the defintion was of any import whatsoever it should have been included in the main body of text, not as an afterthought in the exercises. Over all, though, I think this is a good book for a course in elementary linear algebra.
Rating:
Summary: The clearest and best math text
Review: I've read many great math books in many areas: Real, Complex, Functional, Fourier Analysis, Algebra, Set Theory, etc...
This book is the best.
The proofs are clear, the line of development is thought out, the examples are relevant.
One of the best aspects of the book is in the homework exercises. All sections start off with a list of true/false questions. Answering each question with a proof or counterexample was how I learned to be a clear-thinking math student.
Rating:
Summary: A Space Odyssey
Review: If you major in engineering or economics
please don't buy this one. this is for those
who want to be mathmaticians, or for those
who feel real interest in math. so you guys who think that knowing how to
multiply Matrix and solve System of equations,
please look for the easy, less rigorous one. this might be a little challenging for the first
course text in linear algebra, (actually I feel
difficulty too. when I was taking the course.)
but keep going on and by the time you understand
the first two chapters thourouhly, you'll see why
"Linear Algebra"is said to be one of the most
beautiful part of math. It's not about Multiplying Matrixes... checking
determinant bla bla...
It's about the space and the relation between them. peace.
Rating:
Summary: The Best I've Seen
Review: In my experience thus far, I have not come read another introductory algebra that is as comprehensive and thorough as this one. It does not sacrific clarity for the mathematics and similarly it does not sacrifice mathematics for clarity. From the beginning it builds and expects you to keep up as it introduces new topics. There is a definate succession and continuity in this volume which does not exists in many other introductory algebra texts. Furthermore, it presents good proofs and asks for the reader's help where appropriate. The only aspect of the book that I would critique is its problems. Even though they have somewhat challenging ones, there are none which truly test the depths of ones thinking on the material presented. For example, Spivak does this well in his "Calculus". Nonetheless, this is a great book. It covers standard topics with a few applications thrown in for good measure. Even so, it is unmistakebly a math book, not a science/engineering text on mathematics. I would recommend this to anyone who want a solid start in linear algebra.
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