# 抽象代数讲义 第2卷

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`　　The present volume is the second in the author's series of three dealing with abstract algebra.For an understanding of this volume a certain familiarity with the basic concepts treated in Volume I：groups,rings,fields,homomorphisms,is presupposed.However,we have tried to make this account of linear algebra independent of a detailed knowledge of our first volume.References to specific results are given occasionally but some of the fundamental concepts needed have been treated again.In short,it is hoped that this volume can be read with complete understanding by any'student who is mathematically sufficiently mature and who has a familiarity with the standard notions of modern algebra.`

`CHAPTER I：FINITE DIMENSIONA VECTOR SPACES1.Abstract vector spaces2.Right vector spaces3.o-modules4.Linear dependence5.Invariance of dimensionality6.Bases and matrices7.Applications to matrix theory8.Rank of a set of vectors9.Factor spaces10.Algebra of subspaces11.Independent subspaces， direct sumsCHAPTER II：LINEAR TRANSFORMATIONS1.Definition and examples2.Compositions of linear transformations3.The matrix of a linear transformation4.Compositions of matrices5.Change of basis.Equivalence and similarity of matrices6.Rank space and null space of a linear transformation7.Systems of linear equations8.Linear transformations in right vector spaces9.Linear functions10.Duality between a finite dimensional space and itsconjugate space11.Transpose of a linear transformation12.Matrices of the transpose13.ProjectionsCHAPTER III：THE THEORY OF A SINGLE LINEAR TRANSFORMATION1.The minimum polynomial of a linear transformation2.Cyclic subspaces3.Existence of a vector whose order is the minimum polynomial4.Cyclic linear transformations5.The []-module determined by a linear transformation6.Finitely generated o-modules， o， a principal ideal domain7.Normalization of the generators of F and of8.Equivalence of matrices with elements in a principal ideal domain9.Structure of finitely generated a-modules10.Invariance theorems11.Decomposition of a vector space relative to a linear transformation12.The characteristic and minimum polynomials13.Direct proof of Theorem 1314.Formal properties of the trace and the characteristic polynomial15.The ring of a-endomorphisms of a cyclic o-module16.Determination of the ring of a-endomorphisms of a finitely generated o module， o principal17.The linear transformations which commute with a given linear transformation18.The center of the ringCHAPTER IV：SETS OF LINEAR TRANSFORMATIONS1.Invariant subspaces2.Induced linear transformations3.Composition series4.Decomposability5.Complete reducibility6.Relation to the theory of operator groups and the theory of modules7.Reducibility， decomposability， complete reducibility for a single linear transformation8.The primary components of a space relative to a linear transformation9.Sets of commutative linear transformationsCHAPTER V：BILINEAR FORMS1.Bilinear forms2.Matrices of a bilinear form3.Non-degenerate forms4.Transpose of a linear transformation relative to a pair of bilinear forms5.Another relation between linear transformations and bilinear forms6.Scalar products7.Hermitian scalar products8.Matrices of hermitian scalar products9.Symmetric and hermitian scalar products over special division rings10.Alternate scalar products11.Witt''s theorem12.Non-alternate skew-symmetric formsCHAPTER VI：EUCLIDEAN AND UNITARY SPACES1.Cartesian bases2.Linear transformations and scalar products3.Orthogonal complete reducibility4.Symmetric， skew and orthogonal linear transformations5.Canonical matrices for symmetric and skew linear transformations6.Commutative symmetric and skew linear transformations7.Normal and orthogonal linear transformations8.Semi-definite transformations9.Polar factorization of an arbitrary linear transformation10.Unitary geometry11.Analytic functions of linear transformationsCHAPTER VII：PRODUCTS OF VECTOR SPACES1.Product groups of vector spaces2.Direct products of linear transformations3.Two-sided vector spaces4.The Kronecker product5.Kronecker products of linear transformations and of matrices6.Tensor spaces7.Symmetry classes of tensors8.Extension of the field of a vector space9.A theorem on similarity of sets of matrices10.Alternative definition of an algebra.Kronecker product of algebrasCHAPTER viii：THE RING OF LINEAR TRANSFORMATIONS1.Simplicity of2.Operator methods3.The left ideals of4.Right ideals5.Isomorphisms of rings of linear transformationsCHAPTER IX：INFINITE DIMENSIONAL VECTOR SPACES1.Existence of a basis2.Invariance of dimensionality3.Subspaces4.Linear transformations and matrices5.Dimensionality of the conjugate space6.Finite topology for linear transformations7.Total subspaces of R*8.Dual spaces.Kronecker products9.Two-sided ideals in the ring of linear transformations10.Dense rings of linear transformations11.Isomorphism theorems12.Anti-automorphisms and scalar products13.Schur''s lemma.A general density theorem14.Irreducible algebras of linear transformationsIndex`

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