标题：高等线性代数
图书信息
出版社: 世界图书出版公司; 第3版 (2008年8月1日)
外文书名: Advanced Linear Algebra
平装: 522页
正文语种: 英语
开本: 24
ISBN: 9787506292528
条形码: 9787506292528
尺寸: 22.4 x 14.8 x 2.4 cm
重量: 680 g
作者简介
作者：(美国)罗曼 (Roman.S)
内容简介
《高等线性代数(第3版)》is a thorough introduction to linear algebra，for the graduate or advanced undergraduate student。 Prerequisites are limited to a knowledge of the basic properties of matrices and determinants。 However，since we cover the basics of vector spaces and linear transformations rather rapidly，a prior course in linear algebra （even at the sophomore level），along with a certain measure of "mathematical maturity," is highly desirable。
目录
Preface to the Third Edition，vii
Preface to the Second Edition，ix
Preface to the First Edition，xi
Preliminaries
Part 1: Preliminaries
Part 2: Algebraic Structures
Part I-Basic Linear Algebra
1 Vector Spaces
Vector Spaces
Subspaces
Direct Sums
Spanning Sets and Linear Independence
The Dimension of a Vector Space
Ordered Bases and Coordinate Matrices
The Row and Column Spaces of a Matrix
The C0mplexification of a Real Vector Space
Exercises
2 Linear Transformations
Linear Transformations
The Kernel and Image of a Linear Transformation
Isomorphisms
The Rank Plus Nullity Theorem
Linear Transformations from Fn to Fm
Change of Basis Matrices
The Matrix of a Linear Transformation
Change of Bases for Linear Transformations
Equivalence of Matrices
Similarity of Matrices
Similarity of Operators
Invariant Subspaces and Reducing Pairs
Projection Operators
Topological Vector Spaces
Linear Operators on Vc
Exercises
3 The Isomorphism Theorems
Quotient Spaces
The Universal Property of Quotients and the First Isomorphism Theorem
Quotient Spaces，Complements and Codimension
Additional Isomorphism Theorems
Linear Functionals
Dual Bases
Reflexivity
Annihilators
Operator Adjoints
Exercises
4 Modules I: Basic Properties
Motivation
Modules
Submodules
Spanning Sets
Linear Independence
Torsion Elements
Annihilators
Free Modules
Homomorphisms
Quotient Modules
The Correspondence and Isomorphism Theorems
Direct Sums and Direct Summands
Modules Are Not as Nice as Vector Spaces
Exercises
5 Modules II: Free and Noetherian Modules
The Rank of a Free Module
Free Modules and Epimorphisms
Noetherian Modules
The Hilbert Basis Theorem
Exercises
6 Modules over a Principal Ideal Domain
Annihilators and Orders
Cyclic Modules
Free Modules over a Principal Ideal Domain
Torsion-Free and Free Modules
The Primary Cyclic Decomposition Theorem
The Invariant Factor Decomposition
Characterizing Cyclic Modules
lndecomposable Modules
Exercises
Indecomposable Modules 
Exercises 159
7 The Structure of a Linear Operator 
The Module Associated with a Linear Operator 
The Primary Cyclic Decomposition of VT 
The Characteristic Polynomial 
Cyclic and Indecomposable Modules
The Big Picture 
The Rational Canonical Form 
Exercises
8 Eigenvalues and Eigenvectors 
Eigenvalues and Eigenvectors 
Geometric and Algebraic Multiplicities 
The Jordan Canonical Form 
Triangularizability and Schur's Theorem 
Diagonalizable Operators 
Exercises
9 Real and Complex Inner Product Spaces 
Norm and Distance 
Isometrics
Orthogonality 
Orthogonal and Orthonormal Sets 
The Projection Theorem and Best Approximations 
The Riesz Representation Theorem 
Exercises
10 Structure Theory for Normal Operators 
The Adjoint of a Linear Operator 
Orthogonal Projections 
Unitary Diagonalizability 
Normal Operators 
Special Types of Normal Operators 
Seif-Adjoint Operators 
Unitary Operators and Isometries 
The Structure of Normal Operators 
Functional Calculus 
Positive Operators 
The Polar Decomposition of an Operator 
Exercises
Part Ⅱ-Topics
11 Metric Vector Spaces: The Theory of Bilinear Forms 
Symmetric Skew-Symmetric and Alternate Forms 
The Matrix ofa Bilinear Form 
Quadratic Forms 
Orthogonality 
Linear Functionals 
Orthogonal Complements and Orthogonal Direct Sums 
Isometrics 
Hyperbolic Spaces 
Nonsingular Completions ofa Subspace 
The Witt Theorems: A Preview 
The Classification Problem for Metric Vector Spaces 
Symplectic Geometry 
The Structure of Orthogonal Geometries: Orthogonal Bases 
The Classification of Orthogonal Geometries:Canonical Forms 
The Orthogonal Group 
The Witt Theorems for Orthogonal Geometries 
Maximal Hyperbolic Subspaces of an Orthogonal Geometry 
Exercises
12 Metric Spaces 
The Definition 
Open and Closed Sets 
Convergence in a Metric Space 
The Closure of a Set 
Dense Subsets 
Continuity 
Completeness 
Isometrics 
The Completion of a Metric Space 
Exercises
13 Hilbert Spaces 
A Brief Review 
Hilbert Spaces 
Infinite Series 
An Approximation Problem 
Hilbert Bases 
Fourier Expansions 
A Characterization of Hilbert Bases 
Hilbert Dimension 
A Characterization of Hilbert Spaces 
The Riesz Representation Theorem 
Exercises
14 Tensor Products 
Universality 
Bilinear Maps 
Tensor Products 
When Is a Tensor Product Zero? 
Coordinate Matrices and Rank 
Characterizing Vectors in a Tensor Product 
Defining Linear Transformations on a Tensor Product 
The Tensor Product of Linear Transformations 
Change of Base Field 
Multilinear Maps and Iterated Tensor Products 
Tensor Spaces 
Special Multilinear Maps 
Graded Algebras 
The Symmetric and Antisymmetric Tensor Algebras 
The Determinant 
Exercises
15 Positive Solutions to Linear Systems:Convexity and Separation 
Convex Closed and Compact Sets 
Convex Hulls 
Linear and Affine Hyperplanes 
Separation 
Exercises
16 Affine Geometry 
Affine Geometry 
Affine Combinations 
Affine Hulls 
The Lattice of Flats 
Affine Independence 
Affine Transformations 
Projective Geometry 
Exercises
17 Singular Values and the Moore-Penrose Inverse 
Singular Values 
The Moore-Penrose Generalized Inverse 
Least Squares Approximation 
Exercises
18 An Introduction to Algebras 
Motivation 
Associative Algebras 
Division Algebras 
Exercises
19 The Umbral Calculus 
Formal Power Series 
The Umbral Algebra 
Formal Power Series as Linear Operators 
Sheffer Sequences 
Examples of Sheffer Sequences 
Umbral Operators and Umbral Shifts 
Continuous Operators on the Umbral Algebra 
Operator Adjoints 
Umbral Operators and Automorphisms of the Umbral Algebra 
Umbral Shifts and Derivations of the Umbral Algebra 
The Transfer Formulas 
A Final Remark 
Exercises
References
Index of Symbols
Index