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Gilbert Strang is Brilliant. The book is a treasure trove of his insight into the subtleties of Linear Algebra. As another reviewer pointed out, do the review problems. If you do them in sequence they tend to help clarify the content of the chapters, and prompt you into thinking about the material in different ways. The Problems teach as much as the sections explaining them.
A good book influences its reader in an unimpressive manner. This is a book initiates the reader with the right seeds of the subject.