线性代数导论 第2版（影印版）

　　 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 im mediately after, the first course in calculus.
　　

chapter i
vectors
1. definition of points in space
2. located vectors
3. scalar product
4. the norm of a vector
5. parametric lines
6. planes
chapter ii
matrices and linear equations
1. matrices
2. multiplication of matrices
3. homogeneous linear equations and elimination
4. row operations and gauss elimination
5 row operations and elementary matrices
6. linear combinations
chapter iii
vector spaces
1. definitions
2. linear combinations

　　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 im mediately 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 under standing 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 interpre tations. 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.
　　 The first chapter is designed to serve several purposes. First, and most basically, it establishes the fundamental connection between linear algebra and geometric intuition. There are indeed two aspects (at least) to linear algebra: the formal manipulative aspect of computations with matrices, and the geometric interpretation. I do not wish to prejudice one in favor of the other, and I believe that grounding formal manipulations in geometric contexts gives a very valuable background for those who use linear algebra. Second, this first chapter gives immediately concrete examples, with coordinates, for linear combinations, perpendicu larity, and other notions developed later in the book. In addition to the geometric context, discussion of these notions provides examples for subspaces, and also gives a fundamental interpretation for linear equations. Thus the first chapter gives a quick overview of many topics in the book. The content of the first chapter is also the most fundamental part of what is used in calculus courses concerning functions of several variables, which can do a lot of things without the more general ma-trices. If students have covered the material of Chapter I in another course, or if the instructor wishes to emphasize matrices right away, then the first chapter can be skipped, or can be used selectively for examples and motivation.
　　 After this introductory chapter, we start with linear equations, matrices, and Gauss elimination. This chapter emphasizes computational aspects of linear algebra. Then we deal with vector spaces, linear maps and scalar products, and their relations to matrices. This mixes both the computational and theoretical aspects.
　　 Determinants are treated much more briefly than in the first edition, and several proofs are omitted. Students interested in theory can refer to a more complete treatment in theoretical books on linear algebra.
　　 I have included a chapter on eigenvalues and eigenvectors. This gives practice for notions studied previously, and leads into material which is used constantly in all parts of mathematics and its applications.
　　 I am much indebted to Toby Orloff and Daniel Horn for their useful comments and corrections as they were teaching the course from a preliminary version of this book. I thank Allen Altman and Gimli Khazad for lists of corrections.