高等線性代數(第3版)（簡體書）

This book 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。本書是Springer《數學研究生教材》第135卷．全書分為兩部分，第一部分主要闡述線性代數基本理論，第二部分對九個專題做了討論．
目次：（一）線性代數基本理論：矢量空間；線性變換；同構定理；模；通過理想整環建立的模，線性算子結構；特徵值和特徵向量；實與複內積空間；譜定理用於正規算子。（二）專題：度量向量空間；度量空間，希耳伯特空間；張量乘積仿射幾何學；啞運算．
讀者對象：數學及有關專業高年級大學生和研究生．
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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
　7　The Structure of a Linear Operator
　8　Eigenvalues and Eigenvectors
　9　Real and Complex Inner Product Spaces
　10 Structure Theory for Normal Operators
Part ll--Topics
　11 Metric Vector Spaces: The Theory of Bilinear Forms
　12 Metric Spaces
　13 Hilbert Spaces
　14 Tensor Products
　15 Positive Solutions to Linear Systems:Convexity and Separation
　16 Affine Geometry
　17 Singular Values and the Moore-Penrose Inverse
　18 An Introduction to Algebras
　19 The Umbral Calculus
Referenees
Index of Symbols
Index