5分
高等线性代数-(第3版)
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。
- 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 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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