《抽象代数讲义》是一套久负盛名的三卷集教材，是作者雅格布斯根据他在霍普金斯大学和耶鲁大学讲课时的讲义编写而成的，后又成为作者《基本代数学》一书的蓝本。第1卷介绍了群、环、域、同构等抽象代数的重要的基本概念和抽象代数的基本性质。《抽象代数讲义（第2卷》主要涉及线性代数理论，着重论述了向量空间理论。

CHAPTER I: FINITE DIMENSIONAL VECTOR SPACES

SECTION

 1.Abstract vector spaces

 2.Rightvectorspaces

 3.o-modules

 4.Linear dependence

 5.Invariance of dimensionality

 6.Bases and matrices

 7.Applications to matrix theory

 8.Rank ofa set ofvectors

 9.Factor spaces

 10.Algebra ofsubspaces

 11.Independent subspaces, direct sums

CHAPTER II: LINEAR TRANSFORMATIONS

 1.Definition and examples

 2.Compositions of linear transformations

 3.The matrix of a linear transformation

 4.Compositions ofmatrices

 5.Change of basis.Equivalence and similarity of matrices

 6.Rank space and null space of a linear transformation

 7.Systems oflinear equations

 8.Linear transformations in right vector spaces

 9.Linear functions

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

 11.Transpose of a linear transformation

 12.Matrices of the transpose

 13.Projections

CHAPTER III: THE THEORY OF A SINGLE LINEAR TRANSFORMATION

 1.The minimum polynomial of a linear transformation

 2.Cyclicsubspaces

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

 4.Cyclic linear transformations

 5.The module det:ermiried by a linear transformation

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

 7.Normalization of the generators of; and of

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

 9.Structure of finitely generated o-modules

 10.Invarjance theorems

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

 12.The characteristic and minimum polynomials

 13.Direct proof of Theorem 13

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

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

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

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

 18.The center of the ring

CHAPTER Ⅳ: SETS OF LINEAR TRANSFORMATIONS

 1.Invariant subspaces

 2.Induced linear transformations

 3.Composition series

 4.Decomposability

 5.Complete reducibility

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

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

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

 9.Sets of commutative linear transformations

CHAPTER Ⅴ: BILINEAR FORMS

 1.Bilinear forms

 2.Matrices of a bilinear form

SECTION

 3.Non-degenerate forms

 4.Transpose of a linear transformation relative to a pair-of bilinear forms

 5.Another relation between linear transformations and bilinear forms

 6.Scalar products

 7.Hermitian scalar products

 8.Matrices of hermitian scalar products

 9.Symmetric and hermitian scalar products over special division rings

 10.Alternate scalar products

 11.Witt's theorem

 12.Non-alternate skew-symmetric forms

CHAPTER VI; EUCLIDEAN" AND UNITARY SPACES

 1.Cartesian bases

 2.Linear transformations and scalar products

 3.Orthogonal complete reducibility

 4.Symmetric, skew and orthogonal linear transformations

 5.Canonical matrices for symmetric and skew linear transformations

 6.Commutative symmetric and skew ]{near transformations

 7.Normal and orthogonal linear transformations

 8.Semi-definite transformations

 9.Polar factorization of an arbitrary ]{near transformation

 10.Unitary geometry

 11.Analytic functions of linear transformations

CHAPTER VII PRODUCTS OF VECTOR SPA(~ES

 1.PrOduct groups of vector spaces

 2.Direct products of linear transformations

 3.Two-sided vector spaces

 4.The Kronecker product

 5.Kronecker products of linear transformations and of matrices

 6.Tensor spaces

 7.Symmetry classes of tensors

 8.Extension of the field of a vector space

 9.A theorem on similarity of sets of matrices

SECTION

 10.A_Iternativc definition of an algebra.Kronecker product of algebras

CHAPTER VIII: THE RING OF LINEAR TRANSFORMATIONS

 1.Simplicity of

 2.Operator methods

 3.The left ideals of

 4.Right ideals

 5.Isomorphisms of rings of linear transformations

CHAPTER IX: INFINITE DIMENSIONAL VECTOR SPACES

 1.Existence of a basis

 2.Invariance of dimensionality

 3.Subspaces

 4.Linear transformations and matrices

 5.Dimensionality of the conjugate space

 6.Finite topology for linear transformations

 7.Total subspaces of 9~*

 8.Dual spaces.Kronecker products

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

 10.Dense rings of linear transformations

 11.Isomorphism theorems

 12.Anti-automorphisms and scalar products

 13.Schur's lemma.A general density theorem

 14.Irreducible algebras of linear transformations

Index