Linear algebra

목차 (Table of Contents)

• CONTENTS
• CHAPTER Ⅰ Vectors in $$R^n$$
• 1. Definition of points in n-space = 1
• 2. Located vectors = 4
• 3. Scalar product = 6
• 4. The norm of a vector = 8
• 5. Lines and planes = 13
• 6. Complex numbers = 17
• CHAPTER Ⅱ Vector Spaces
• 1. Terminology = 21
• 2. Definitions = 21
• 3. Bases = 26
• 4. Dimension of a vector space = 31
• 5. Sums and direct sums = 34
• CHAPTER Ⅲ Matrices
• 1. The space of matrices = 37
• 2. Linear equations = 42
• 3. Multiplication of matrices = 45
• Appendix. Elimination = 52
• CHAPTER Ⅳ Linear Mappings
• 1. Mappings = 55
• 2. Linear mappings = 60
• 3. The kernel and image of a linear map = 67
• 4. Dimensions of kernel and image = 69
• 5. Composition of linear mappings = 74
• CHAPTER Ⅴ Linear Maps and Matrices
• 1. The linear map associated with a matrix = 78
• 2. The matrix associated with a linear map = 80
• 3. Composition of linear mappings = 85
• CHAPTER Ⅵ Determinants
• 1. Determinants of order 2 = 89
• 2. Properties of determinants = 90
• 3. Cramer's rule = 93
• 4. Existence of determinants = 96
• 5. Permutations = 102
• 6. Uniqueness = 106
• 7. Determinant of a transpose = 109
• 8. Determinant of a product = 110
• 9. Inverse of a matrix = 112
• 10. The determinant of a linear map = 113
• CHAPTER Ⅶ Scalar Products and Orthogonality
• 1. Scalar products = 116
• 2. Positive definite products = 118
• 3. General orthogonal bases = 128
• 4. The dual space = 130
• 5. The rank of a matrix and linear equations = 137
• CHAPTER Ⅷ Matrices and Bilinear Maps
• 1. Bilinear forms = 142
• 2. Quadratic forms = 148
• 3. Symmetric operators = 150
• 4. Hermitian operators = 154
• 5. Unitary operators = 159
• 6. Sylvester's theorem = 162
• CHAPTER Ⅸ Polynomials and Matrices
• 1. Polynomials = 167
• 2. Polynomials of matrices and linear maps = 170
• 3. Eigenvectors and eigenvalues = 172
• 4. The characteristic polynomial = 178
• CHAPTER Ⅹ Triangulation of Matrices and Linear Maps
• 1. Existence of triangulation = 183
• 2. Theorem of Hamilton-Cayley = 186
• 3. Diagonalization of unitary maps = 188
• CHAPTER XI Spectral Theorem
• 1. Eigenvectors of symmetric linear maps = 190
• 2. The spectral theorem = 193
• 3. The complex case = 198
• CHAPTER XII Polynomials and Primary Decompositions
• 1. The Euclidcan algorithm = 200
• 2. Greatest common divisor = 202
• 3. I'niquc factorization = 205
• 4. The integers = 209
• 5. Application to the decomposition of a vector space = 211
• 6. Schur's lemma = 214
• 7. Adic expansion of a polynomial = 216
• CHAPTER XIII Multilinear Products
• 1. The tensor product = 221
• 2. Isomorphisms of tensor products = 225
• 3. Alternating products : Special case = 227
• 4. Alternating products : General case = 231
• Appendix. The vector space generated by a set = 240
• CHAPTER XIV Groups
• 1. Groups and examples = 242
• 2. Simple properties of groups = 245
• 3. Cosets and normal subgroups = 251
• 4. Cyclic groups = 256
• 5. Free abelian groups = 259
• CHAPTER XV Rings
• 1. Rings and ideals = 263
• 2. Homomorphisms = 268
• 3. Modules = 271
• 4. Factor modules = 275
• APPENDIX 1 Convex Sets
• 1. Definition = 279
• 2. Separating hyperplanes = 281
• 3. Extreme points and supporting hyperplanes = 283
• 4. The Krein-Miiman theorem = 285
• APPENDIX 2 Odds and Ends
• 1. Induction = 287
• 2. Algebraic closure of the complex numbers = 288
• 3. Equivalence relations = 289
• Index = 292