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.
Author(s): Serge Lang
Series: Undergraduate Texts in Mathematics
Edition: 2
Publisher: Springer
Year: 1985
Language: English
Pages: 303
Preface......Page 6
Contents......Page 8
§1. Definition of Points in Space......Page 10
§2. Located Vectors......Page 18
§3. Scalar Product......Page 21
§4. The Norm of a Vector......Page 24
§5. Parametric Lines......Page 39
§6. Planes......Page 43
CHAPTER II Matrices and Linear Equations......Page 51
§1. Matrices......Page 52
§2. Multiplication of Matrices......Page 56
§3. Homogeneous Linear Equations and Elimination......Page 73
§4. Row Operations and Gauss Elimination......Page 79
§5. Row Operations and Elementary Matrices......Page 86
§6. Linear Combinations......Page 94
§1. Definitions......Page 97
§2. Linear Combinations......Page 102
§3. Convex Sets......Page 108
§4. Linear Independence......Page 113
§5. Dimension......Page 119
§6. The Rank of a Matrix......Page 124
§1. Mappings......Page 132
§2. Linear Mappings......Page 136
§3. The Kernel and Image of a Linear Map......Page 145
§4. The Rank and Linear Equations Again......Page 153
§5. The Matrix Associated with a Linear Map......Page 159
Appendix: Change of Bases......Page 163
§1. Composition of Linear Maps......Page 167
§2. Inverses......Page 173
§1. Scalar Products......Page 180
§2. Orthogonal Bases......Page 189
§3. Bilinear Maps and Matrices......Page 199
§1. Determinants of Order 2......Page 204
§2. 3 x 3 and n x n Determinants......Page 209
§3. The Rank of a Matrix and Subdeterminants......Page 219
§4. Cramer's Rule......Page 223
§5. Inverse of a Matrix......Page 226
§6. Determinants as Area and Volume......Page 230
§1. Eigenvectors and Eigenvalues......Page 242
§2. The Characteristic Polynomial......Page 247
§3. Eigenvalues and Eigenvectors of Symmetric Matrices......Page 259
§4. Diagonalization of a Symmetric Linear Map......Page 264
Appendix. Complex Numbers......Page 269
Answers to Exercises......Page 274
Index......Page 300
