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Chapter Ⅰ．Linear spaces 1

1．The axioms of a linear space 1

2．Linear subspaces 7

3．Linear spaces of finite dimension 10

Chapter Ⅱ．Linear transformations 16

1．Linear mappings 16

2．Dual spaces 20

3．Dual mappings 24

4．Sum and product of linear mappings 26

Chapter Ⅲ．Matrices 31

1．Matrices and systems of linear equations 31

2．Multiplication of matrices 37

3．Basis-transformation 40

4．Elementary transformations 43

Chapter Ⅳ．Determinants 46

1．Determinant-functions 46

2．The determinant of an endomorphism 50

3．The determinant of a matrix 53

4．Cofactors 57

5．The characteristic polynomial 62

6．The trace 67

Chapter Ⅴ．Oriented linear spaces 71

1．Orientation by a determinant-function 71

2．The topology of a linear space 74

Chapter Ⅵ．Multilinear mappings 81

1．Basic properties 81

2．Tensor-product 85

Chapter Ⅶ．Tensor-algebra 96

1．Tensors 96

2．Contraction 104

3．The duality of Tpq(E) and Tqp(E) 108

Chapter Ⅷ．Exterior algebra 112

1．Covariant skew-symmetric tensors 112

2．Decomposable skew-symmetric tensors 121

3．Mixed skew-symmetric tensors 125

4．The duality of Spq(E) and Sqp(E) 128

Chapter Ⅸ．Duality in exterior algebra 133

1．The dual product 133

2．The tensors Jp 137

3．The dual isomorphisms 142

4．The adjoint tensor 148

5．The exterior product 157

Chapter Ⅹ．Inner product spaces 160

1．The inner product 160

2．Orthonormal bases 164

3．Normed determinant-functions 168

4．The duality in an inner product space 175

5．Normed linear spaces 182

Chapter Ⅺ．Linear mappings of inner product spaces 186

1．The adjoint mapping 186

2．Selfadjoint mappings 189

3．Orthogonal projections 193

4．Skew mappings 196

5．Isometric mappings 201

6．Rotations of the plane and 3-space 205

7．Differentiable families of automorphisms 210

Chapter Ⅻ．Symmetric bilinear functions 220

1．Bilinear and quadratic functions 221

2．The decomposition of E 224

3．Simultaneous diagonalization of two bilinear forms 231

4．Pseudo-Euclidean spaces 237

5．Linear mappings of pseudo-Euclidean spaces 244

Chapter ⅩⅢ．Quadrics 250

1．Affine spaces 250

2．Quadrics in the affine space 255

3．Affine equivalence of quadrics 263

4．Quadrics in the Euclidean space 268

Chapter ⅩⅣ．Unitary spaces 275

1．Hermitian functions 275

2．Unitary spaces 277

3．Linear mappings of a unitary space 281

4．Unitary mappings of the complex plane 286

5．Applications to the orthogonal group 291

6．Application to Lorentz-transformations 297

Chapter ⅩⅤ．Invariant subspaces 302

1．The algebra of polynomials 303

2．Polynomials of endomorphisms 307

3．Invariant subspaces 313

4．The decomposition of a complex linear space 317

5．The decomposition of a real linear space 321

6．Application to inner product spaces 326

Bibliography 333

Subject Index 335