《LINEAR ALGEBRA SECOND EDITION》

Chapter Ⅰ．Linear spaces1

1．The axioms of a linear space1

2．Linear subspaces7

3．Linear spaces of finite dimension10

Chapter Ⅱ．Linear transformations16

1．Linear mappings16

2．Dual spaces20

3．Dual mappings24

4．Sum and product of linear mappings26

Chapter Ⅲ．Matrices31

1．Matrices and systems of linear equations31

2．Multiplication of matrices37

3．Basis-transformation40

4．Elementary transformations43

Chapter Ⅳ．Determinants46

1．Determinant-functions46

2．The determinant of an endomorphism50

3．The determinant of a matrix53

4．Cofactors57

5．The characteristic polynomial62

6．The trace67

Chapter Ⅴ．Oriented linear spaces71

1．Orientation by a determinant-function71

2．The topology of a linear space74

Chapter Ⅵ．Multilinear mappings81

1．Basic properties81

2．Tensor-product85

Chapter Ⅶ．Tensor-algebra96

1．Tensors96

2．Contraction104

3．The duality of Tpq(E) and Tqp(E)108

Chapter Ⅷ．Exterior algebra112

1．Covariant skew-symmetric tensors112

2．Decomposable skew-symmetric tensors121

3．Mixed skew-symmetric tensors125

4．The duality of Spq(E) and Sqp(E)128

Chapter Ⅸ．Duality in exterior algebra133

1．The dual product133

2．The tensors Jp137

3．The dual isomorphisms142

4．The adjoint tensor148

5．The exterior product157

Chapter Ⅹ．Inner product spaces160

1．The inner product160

2．Orthonormal bases164

3．Normed determinant-functions168

4．The duality in an inner product space175

5．Normed linear spaces182

Chapter Ⅺ．Linear mappings of inner product spaces186

1．The adjoint mapping186

2．Selfadjoint mappings189

3．Orthogonal projections193

4．Skew mappings196

5．Isometric mappings201

6．Rotations of the plane and 3-space205

7．Differentiable families of automorphisms210

Chapter Ⅻ．Symmetric bilinear functions220

1．Bilinear and quadratic functions221

2．The decomposition of E224

3．Simultaneous diagonalization of two bilinear forms231

4．Pseudo-Euclidean spaces237

5．Linear mappings of pseudo-Euclidean spaces244

Chapter ⅩⅢ．Quadrics250

1．Affine spaces250

2．Quadrics in the affine space255

3．Affine equivalence of quadrics263

4．Quadrics in the Euclidean space268

Chapter ⅩⅣ．Unitary spaces275

1．Hermitian functions275

2．Unitary spaces277

3．Linear mappings of a unitary space281

4．Unitary mappings of the complex plane286

5．Applications to the orthogonal group291

6．Application to Lorentz-transformations297

Chapter ⅩⅤ．Invariant subspaces302

1．The algebra of polynomials303

2．Polynomials of endomorphisms307

3．Invariant subspaces313

4．The decomposition of a complex linear space317

5．The decomposition of a real linear space321

6．Application to inner product spaces326

Bibliography333

Subject Index335