I'm familiar with three linear algebra textbooks: Gilbert Strang's Linear Algebra and Its Applications, Georgi E. Shilov's Linear Algebra, and now this one. It was recommended to me by one of my brothers, who had the author as a professor at the University of Maryland - College Park.
Gil Strang's book is very well regarded, and I like it, too. However, as a writer, Strang tries a little too hard to be friendly and colloquial. As a result, some of his explanations are less clear than they need to be. It helps that videos of his linear algebra lectures are on the Web at [...], and those lectures clarify some of the "folksy" wording in the textbook. Strang obviously loves his subject and knows it thoroughly, but those qualities, however admirable, do not substitute for clear writing.
Georgi E. Shilov's book is also highly regarded, by me as well. Shilov is one of those no-nonsense Russian mathematicians who's all about the subject and doesn't care if you like him or not. As a result, his writing is very clear and straightforward, albeit a little stiff and formal even in translation. The great virtues of Shilov's book are that the writing is clear and it's very rigorous: in fact, a reader would do well to have some familiarity with abstract algebra before starting it. But the book's virtues are also its weakness: because of the rigorous treatment, Shilov offers considerably less conceptual hand-holding than Strang. Yes, you can understand what he's talking about, but you'd sure better have a strong mathematical background, time, and self-confidence to plow through his book, especially if it's on your own.
Which brings us, finally, to the Lay book. I am delighted to report that Lay combines the informal, encouraging tone and conceptual hand-holding of the Strang book with the clarity of the Shilov book. In other words, they're all good, but for most undergraduates, Lay is the best of the three. There's also an excellent study guide (Linear Algebra and Its Applications: Study Guide (update))for the Lay book.