This text is intended for a one semester introductory course in Linear Algebra at the sophomore level geared toward mathematics majors and motivated students. It was originally extracted from Lang "Linear Algebra," and is now in its second edition (a vast improvement over the first: Lang rarely does the increasingly popular token update).
The text takes a theoretical approach to the subject, and the only applications the reader can expect to see are to other interesting areas of mathematics. With the exception of the last chapter, these are left in the exercises, and Lang does not push them vary far.
The trend in most Linear Algebra texts at this level that attempt to appeal to a large audience (such as engineering students) is away from the Definition-Theorem-Proof approach and towards a less formal presentation based around ideas, discussions as proofs, and applications. I prefer the former approach, which Lang is very much in the tradition of, and believe that the way to teach students how to write rigorous and presentable proofs is by making them read and study them. In fact, I learned how to write proofs from studying this text and working all of Lang's well-chosen exercises.
"Introduction to Linear Algebra" starts at the basics with no prior assumptions on the material the reader knows (the Calculus is used only occasionally in the exercises): the first chapter is on points, vectors, and planes in the Euclidean space, R^n. After that is a chapter introducing matricies, inversion, systems of linear equations, and Gaussian elimination. While the book does spend adequate time on how to perform Gaussian elimination and matrix inversion, it also gives all the proofs that these methods work.
The bulk of the theoretical material comes in Chapters III through V, which respectively present the theories of vector spaces, linear mappings, and composite and inverse mappings. The approach is rigorous, but by no means inaccessible. As is necessary in a course like this, time is spent on establishing clear and solid proofs of basic results that will be treated as almost trivial ("you can show it on your homework to convince yourselves") in more advanced classes - c.f. Lang's "Undergraduate Algebra."
The next two chapters cover scalar products and determinants, and have a somewhat more computational feel to them. There is much theory in the sections on scalar products, but a big focus is also the Gramm-Schmidt method for finding an orthonormal basis. Many of the determinant proofs are in the 2 x 2 and 3 x 3 case to avoid bringing in the full formalism and notation of determinants in general.
The text concludes with what is its most difficult chapter, the one on eigenvectors and eigenvalues. It is the most, however, for applications to physics, and interest applications comprise the last half of the chapter.
If you are ordering this text used, I recommend you take care to find the second edition. The first edition was significantly shorter and covered less material.
This is an introductory text, and not for learning the material that would be included in a second course or part of the algebra sequence at the junior/senior level. For those purposes, I recommend Lang's "Linear Algebra." Portions bear strong (often exact) resemblance to the book at present consideration, but the most basic material is missing and much advanced material is included.
In conclusion, I highly recommend this text for a motivated student who wants a first exposure to Linear Algebra. The text isn't always easy reading, and parts may be a tough climb for readers without much exposure to this type of reading. The experience, however, is well worth it; in mathematics, one really only learns as much as one sweats, so to speak.