Lang S. Introduction to Linear Algebra 2ed 1986 + Sol. Man.

Textbook in PDF format

This book is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues. The book contains a large number of exercises, some of the routine computational type, and others are conceptual.
This book is meant as a short text in linear algebra for a one-term course. Except for an occasional example or exercise the text is logically independent of calculus, and could be taught early. In practice, I expect it to be used mostly for students who have had two or three terms of calculus. The course could also be given simultaneously with, or immediately after, the first course in calculus. I have included some examples concerning vector spaces of functions, but these could be omitted throughout without impairing the understanding of the rest of the book, for those who wish to concentrate exclusively on euclidean space. Furthermore, the reader who does not like n = n can always assume that n = 1, 2, or 3 and omit other interpretations. However, such a reader should note that using n = n simplifies some formulas, say by making them shorter, and should get used to this as rapidly as possible. Furthermore, since one does want to cover both the case n = 2 and n = 3 at the very least, using n to denote either number avoids very tedious repetitions.
Vectors
Matrices and linear Equations
Vector Spaces
Linear Mappings
Composition and Inverse Mappings
Scalar Products and Orthogonality
Determinants
Eigenvectors and Eigenvalues
Answers to Exercises