This book by Lipschutz and Lipson is a new version of a previous Lipschutz-only work. Among the numerous existing books on LA,its quality is hard to beat at any price. It has been much helpful to me, since in the last months, I had to improve my understanding of a handful of elementary and not-so-elementary facts (like Steinitz replacement principle, the implicit equations of a subspace, PLU decomposition, Jordan and rational canonical forms...) and I felt myself kinda lost among my old beloved sources (that include: Hoffmann-Kunze's Linear Algebra (2nd Edition), Gantmacher's two volume set on matrices and Strang's Linear Algebra and Its Applications, as well as several advanced algebra volumes). I decided to update, and I shopped around, spending some money on LA books. Voila, I boughtt Hefferon's Linear Algebra, (excellent, though not what I was looking for), Weinstraub's A Guide to Advanced Linear Algebra (Dolciani Mathematical Expositions) (good on Jordan, but it goes too fast and does not excel on the practical side), Axler's "LA done right" (original, but useless), Lax's Linear Algebra and Its Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts) (awsome, but too advanced to me).
Finally, I bought too additional books that I have found truly satisfactory:
(1) Friedberg-Insel-Spence's Linear Algebra, 4th Edition (a very well crafted and comprehensive introduction to LA, in 400 pages, with a host of interesting examples and problems ; I bought the cheapo EEE edition, maybe one day I will get the hardcover one).
(2) Lipschutz-Lipson's work, that solved several of my darkest doubts. For example, I was amazed while reading their revealing account of cyclic bases associated to nilpotent operators. Simply put, I had never seen this topic explained with such a perfection. Mixing theory, examples, solved problems, and unsolved exercises, you are led to a clear vision of the Jordan canonical form and its intrincacies. Theory meets practice, and the book is rich in easy and illustrative exercises, that serve to fix concepts and techniques. As in other Schaum's books by S. Lipschutz, the reader is faced to the utmost simplicity of main ideas, great clearness and an anti-snobbish style. Lots of educative examples and exercises, with low and medium difficulty... All in all, this book is a JEWEL and an increible bargain. I can't find a similar book of such a didactic value and mathematical rigour, covering a similar scope. You have very good THEORY books, like those I've mentioned, but only one able to clarify what's going on, mixing theory, examples and problems. It's just Lipschutz's book.
Let me mention two more advanced LA Problems-Only books: (A) Proskuriakov's 2000 Problemas de álgebra lineal (English and Spanish Edition)'s (the Spanish translation of the original Russian work, which was published in English by Moscow MIR editors 30 years ago as "Problems in Linear Algebra - I.V. Proskuryakov"); (B) Zhang's Linear Algebra (Johns Hopkins Studies in the Mathematical Sciences), not as comprehensive as Proskuriakov, but very good. To learn a subject, two or three well chosen books are usually more helpful than only one.