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CHAPTERⅠ：FINITE DIMENSIONAL VECTOR SPACES3

1.Abstract vector spaces3

2.Right vector spaces6

3.o-modules7

4.Linear dependence9

5.Invariance of dimensionality13

6.Bases and matrices15

7.Applications to matrix theory18

8.Rank of a set of vectors22

9.Factor spaces25

10.Algebra of subspaces25

11.Independent subspaces， direct sums28

CHAPTER Ⅱ： LINEAR TRANSFORMATIONS31

1.Definition and examples31

2.Compositions of linear transformations33

3.The matrix of a linear transformation36

4.Compositions of matrices38

5.Change of basis.Equivalence and similarity of matrices41

6.Rank space and null space of a linear transformation44

7.Systems of linear equations47

8.Linear transformations in right vector spaces49

9.Linear functions51

10.Duality between a finite dimensional space and its conjugate space53

11.Transpose of a linear transformation56

12.Matrices of the transpose58

13.Projections59

CHAPTER Ⅲ： THE THEORY OF A SINGLE LINEAR TRANSFORMATION63

1.The minimum polynomial of a linear transformation63

2.Cyclic subspaces66

3.Existence of a vector whose order is the minimum polynomial67

4.Cyclic linear transformations69

5.The Φ[λ]-module determined by a linear transformation74

6.Finitely generated o-modules， o， a principal ideal domain76

7.Normalization of the generators of ？ and of ？78

8.Equivalence of matrices with elements in a principal ideal domain79

9.Structure of finitely generated o-modules85

10.Invariance theorems88

11.Decomposition of a vector space relative to a linear trans-formation92

12.The characteristic and minimum polynomials98

13.Direct proof of Theorem 13100

14.Formal properties of the trace and the characteristic poly-nomial103

15.The ring of o-endomorphisms of a cyclic o-module106

16.Determination of the ring of o-endomorphisms of a finitely generated o-module， o principal108

17.The linear transformations which commute with a given lin-ear transformation110

18.The center of the ring ？113

CHAPTER Ⅳ： SETS OF LINEAR TRANSFORMATIONS115

1.Invariant subspaces115

2.Induced linear transformations117

3.Composition series120

4.Decomposability122

5.Complete reducibility124

6.Relation to the theory of operator groups and the theory of modules126

7.Reducibility， decomposability， complete reducibility for a single linear transformation128

8.The primary components of a space relative to a linear trans-formation130

9.Sets of commutative linear transformations132

CHAPTER Ⅴ： BILINEAR FORMS137

1.Bilinear forms137

2.Matrices of a bilinear form138

3.Non-degenerate forms140

4.Transpose of a linear transformation relative to a pair of bi-linear forms142

5.Another relation between linear transformations and bilinear forms145

6.Scalar products147

7.Hermitian scalar products150

8.Matrices of hermitian scalar products152

9.Symmetric and hermitian scalar products over special divi-sion rings154

10.Alternate scalar products159

11.Witt’s theorem162

12.Non-alternate skew-symmetric forms170

CHAPTER Ⅵ： EUCLIDEAN AND UNITARY SPACES173

1.Cartesian bases173

2.Linear transformations and scalar products176

3.Orthogonal complete reducibility177

4.Symmetric， skew and orthogonal linear transformations178

5.Canonical matrices for symmetric and skew linear transfor-mations179

6.Commutative symmetric and skew linear transformations182

7.Normal and orthogonal linear transformations184

8.Semi-definite transformations186

9.Polar factorization of an arbitrary linear transformation188

10.Unitary geometry190

11.Analytic functions of linear transformations194

CHAPTER Ⅶ： PRODUCTS OF VECTOR SPACES199

1.Product groups of vector spaces199

2.Direct products of linear transformations202

3.Two-sided vector spaces204

4.The Kronecker product208

5.Kronecker products of linear transformations and of matrices211

6.Tensor spaces213

7.Symmetry classes of tensors217

8.Extension of the field of a vector space221

9.A theorem on similarity of sets of matrices222

10.Alternative definition of an algebra.Kronecker product of algebras225

CHAPTER Ⅷ： THE RING OF LINEAR TRANSFORMATIONS227

1.Simplicity of ？227

2.Operator methods229

3.The left ideals of ？230

4.Right ideals232

5.Isomorphisms of rings of linear transformations233

CHAPTER Ⅸ： INFINITE DIMENSIONAL VECTOR SPACES239

1.Existence of a basis239

2.Invariance of dimensionality240

3.Subspaces242

4.Linear transformations and matrices243

5.Dimensionality of the conjugate space244

6.Finite topology for linear transformations248

7.Total subspaces of ？251

8.Dual spaces.Kronecker products253

9.Two-sided ideals in the ring of linear transformations256

10.Dense rings of linear transformations259

11.Isomorphism theorems264

12.Anti-automorphisms and scalar products267

13.Schur’s lemma.A general density theorem271

14.Irreducible algebras of linear transformations274

Index277