高等線性代數

《高等線性代數》是一本羅曼編制，由世界圖書出版公司在2008年出版的書籍。

基本介紹

• ISBN：9787506292528
• 頁數：522頁
• 出版社：世界圖書出版公司;
• 開本：24

圖書信息

出版社: 世界圖書出版公司; 第3版 (2008年8月1日)

外文書名: Advanced Linear Algebra

平裝: 522頁

正文語種: 英語

開本: 24

ISBN: 9787506292528

條形碼: 9787506292528

尺寸: 22.4 x 14.8 x 2.4 cm

重量: 680 g

作者簡介

作者：(美國)羅曼 (Roman.S)

內容簡介

《高等線性代數(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。

目錄

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