A First Course in Linear Algebra
I examined this book carefully last semester while searching for a good inexpensive (or free) textbook to adopt for a sophomore-level linear algebra course. This book contains all the topics that I'd normally cover in such a course, plus more. The prose is often conversational, but ultimately accurate, unambiguous and lucid.
The book does have some quirks, the most noticeable of which is an extensive reliance on acronyms. Chapters are not numbered, but rather tagged with sometimes cryptic abbreviations. For example, the book begins with Chapter SLE (Systems of Linear Equations), followed by Chapter V (Vectors), then Chapter M (Matrices), etc. Even theorems, definitions, examples and diagrams are designated in this way. For instance, Definition ROLT is for the rank of a linear transformation. (Why not just call it "Definition RANK"?) All this takes some getting used to, but such brevity may have a place in classroom exposition.
If there is a serious omission, it is that the book has scarcely any figures at all, which is surprising given the geometric nature of linear algebra. And I felt that the occasional figures fell short of really illuminating the ideas that they were supposed to convey. For example, Diagram NILT (non-injective linear transformation) is identical to Diagram NSLT (non-surjective linear transformation), except for labeling. So by themselves they don't clearly differentiate the two ideas. Further, these illustrations show generic point sets, not vector spaces. I'd be more comfortable seeing (say) non-surjectivity illustrated by a map from 2-D space to a plane in 3-D space, etc.
I had trouble locating an index in the on-line version of the book.
I found no mistakes at all.
I believe the book is very up to date. Some instructors may want to see a little more on matrix decompositions, but this is not an issue with me. Regardless, because of the non-numeric labeling of chapters and definitions, it would be very easy for the author to add material without affecting the numbering of subsequent sections. For this reason, I rank the book's longevity as high.
The prose is very clear, and one feels that it has been informed by many years of teaching the subject. As mentioned above, I believe that it would be even clearer with the addition of well-crafted figures.
The author has done an excellent job here. The book is remarkably uniform in tone and format, and is uniquely Beezer's work from beginning to end. He has created his own brand of textbook.
The book is broken into sections and subsections, and theorems, proofs, definitions and examples are clearly delineated. The acronym labeling scheme makes the book feel especially modular, possibly at the expense of emphasizing the interdependency among the various topics.
The sequencing is perfectly logical and natural, and l would see no reason to do anything in a different order. This is one instance where the acronyms seem out of place, as a simple numeric labeling of the chapters would underscore the importance of the flow of ideas in a way that the acronyms do not.
I read the online version, which I thought was pretty good. I did find some aspects of the experience to be slightly disconcerting. For example, it's hard to gauge how long a section will be when clicking on an example can suddenly expand a simple phrase to an entire page, or more. But whatever problems I had may have been due to my own preference for thumbing through paper books.
I found no problems with the grammar.
It is difficult to imagine how linear algebra could be culturally insensitive. At any rate, I can't imagine that the author has offended anyone.