Serge Lang's Introduction to Linear Algebra provides a nice introduction to the subject. The text, which is designed for a one semester course for students who are taking or have completed multi-variable calculus, covers the basic theory and computational techniques. Since the emphasis is on proving theorems rather than the applications that are of interest to physical scientists, engineers, and economists, the text is best suited to pure mathematics students.
Topics are motivated, the theory is carefully developed, computational techniques are demonstrated through clearly written examples, and geometric interpretations of the algebra are discussed. The exposition is generally clear, but I occasionally had to turn to Blyth and Robertson's Basic Linear Algebra 2nd Edition or Friedberg, Insel, and Spence's Linear Algebra (4th Edition) for clarification when examples were lacking (notably in the section on eigenvalues and eigenvectors that precedes the introduction of the characteristic polynomial). Another caveat is that there are also numerous errors, including some in the answer key.
The exercises consist of computational problems, which require meticulous attention to detail, and proofs of results that extend the topics developed in the text. The exercises are organized thematically in order to teach concepts not covered in the body of the text. Some problems are reintroduced after additional material has been developed, so that you can solve them in new, more efficient, ways, thereby demonstrating the power of the new techniques that you are learning. Answers to most of the exercises are provided in an appendix, making the text suitable for self-study.
The text begins with a review of vectors. This material is drawn from Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics) and should be familiar to most readers. Next, Lang demonstrates how matrix algebra can be used to solve systems of linear equations. While the reader presumably learned how to solve systems of linear equations in high school (or even earlier), the discussion of homogeneous linear equations, row operations, and linear combinations provides the foundation for subsequent topics in the book.
The remainder of the book is devoted to finite-dimensional vector spaces. Once Lang introduces the basic definitions, he covers linear independence, the basis of a vector space, and dimension. This leads to a discussion of linear mappings, their representation by matrices, and how the kernel and image of these maps are related to the rank of the matrix of linear transformation. Lang discusses composition of mappings and inverse mappings before delving into scalar products, orthogonal bases, and bilinear maps. Lang then develops the theory of determinants and discusses how to apply them to solving systems of linear equations, finding the inverse of a matrix, and calculating areas and volumes. After introducing eigenvectors, eigenvalues, and the characteristic polynomial, Lang concludes the book with a discussion of the eigenvectors and eigenvalues of symmetric matrices that uses the earlier material on scalar products and orthogonality.
Much of this material is drawn from Lang's Linear Algebra (Undergraduate Texts in Mathematics), where it is treated in more depth. However, that text is written for students already familiar with basic matrix manipulations, so it does not discuss elementary matrices or Gaussian elimination. Understanding it also requires greater mathematical sophistication.
This text is limited in scope. If you are preparing to do graduate work in mathematics, you will need to read an additional text such as Lang's Linear Algebra, Friedberg, Insel, and Spence's Linear Algebra, Hoffman and Kunze's Linear Algebra (2nd Edition), Axler's Linear Algebra Done Right, or Blyth and Robertson's Further Linear Algebra. Of these, the one that is most suitable for self-study is Further Linear Algebra.
If you are interested in an introductory text that is suitable for self-study, you may wish to consider Blyth and Robertson's Basic Linear Algebra as an alternative to this one, as it includes abundant examples and answers to almost all the exercises.