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高等线性代数-(第3版)
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高等线性代数-(第3版)

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研究生阶段较为高阶的代数课程

线性代数是理工科本科必开的数学课,但是本书绝对不是理工科线性代数入门课程之后的第二门课程,起码应该有本科阶段抽象代数的基本训练以后才能研读本书。哪怕作者在序言中,说只需要一般的线性代数知识和“数学的成熟度”。 往往知识学起来不那么困难,多重复几遍,坚韧不拔必有所获,而某个学科的“成熟度”,这个实在是需要一定的领悟和天赋才行。当然,勤能补拙,如果一个领域某本书,反复看反复练习十遍八遍的,一般人也是能搞懂的。 人类目前能比较裕如处理的,可能绝大部分就集中在线性关系上了。哪怕是那些非线性的,往往也是用线性来“逼近”,掌握更高抽象层次的线性代数背后的思路和方法,对做其他领域或多或少会有不同程度的启发意义。

2022-08-25 17:01:41
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图文详情
  • ISBN:9787506292528
  • 装帧:一般胶版纸
  • 册数:暂无
  • 重量:暂无
  • 开本:24开
  • 页数:522
  • 出版时间:2008-08-01
  • 条形码:9787506292528 ; 978-7-5062-9252-8

本书特色

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.

内容简介

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。

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
 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
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