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内容简介

　　The audacious title of this book deserves an explanation. Almost all linear algebra books use determinants to prove that every linear operator on a finite-dimensional complex vector space has an eigenvalue. Determinants are difficult, nonintuitive, and often defined without motivation. To prove the theorem about existence of eigenvalues on complex vector spaces, most books must.define determinants, prove that a linear map is not invertible ff and only if its determinant equals O, and then define the characteristic polynomial. This tortuous (torturous?) path gives students little feeling for why eigenvalues must exist. In contrast, the simple determinant-free proofs presented here offer more insight. Once determinants have been banished to the end of the book, a new route opens to the main goal of linear algebra-- understanding the structure of linear operators.

内页插图

目录

Preface to the Instructor
Preface to the Student
Acknowledgments
CHAPTER 1
Vector Spaces
Complex Numbers
Definition of Vector Space
Properties of Vector Spaces
Subspaces
Sums and Direct Sums
Exercises

CHAPTER 2
Finite-Dimenslonal Vector Spaces
Span and Linear Independence
Bases
Dimension
Exercises

CHAPTER 3
Linear Maps
Definitions and Examples
Null Spaces and Ranges
The Matrix of a Linear Map
Invertibility
Exercises

CHAPTER 4
Potynomiags
Degree
Complex Coefficients
Real Coefflcients
Exercises

CHAPTER 5
Eigenvalues and Eigenvectors
lnvariant Subspaces
Polynomials Applied to Operators
Upper-Triangular Matrices
Diagonal Matrices
Invariant Subspaces on Real Vector Spaces
Exercises

CHAPTER 6
Inner-Product spaces
Inner Products
Norms
Orthonormal Bases
Orthogonal Projections and Minimization Problems
Linear Functionals and Adjoints
Exercises

CHAPTER 7
Operators on Inner-Product Spaces
Self-Adjoint and Normal Operators
The Spectral Theorem

Normal Operators on Real Inner-Product Spaces
Positive Operators
Isometries
Polar and Singular-Value Decompositions
Exercises

CHAPTER 8
Operators on Complex Vector Spaces
Generalized Eigenvectors
The Characteristic Polynomial
Decomposition of an Operator
Square Roots
The Minimal Polynomial
Jordan Form
Exercises

CHAPTER 9
Operators on Real Vector Spaces
Eigenvalues of Square Matrices
Block Upper-Triangular Matrices
The Characteristic Polynomial
Exercises

CHAPTER 10
Trace and Determinant
Change of Basis
Trace
Determinant of an Operator
Determinant of a Matrix
Volume
Exercises
Symbol Index
Index

前言/序言

　　You are probably about to teach a course that will give students their second exposure to linear algebra. During their first brush with the subject， your students probably worked with Euclidean spaces and matrices. In contrast， this course will emphasize abstract vector spaces and linear maps.
　　The audacious title of this book deserves an explanation. Almost all linear algebra books use determinants to prove that every linear operator on a finite-dimensional complex vector space has an eigenvalue.Determinants are difficult， nonintuitive， and often defined without motivation. To prove the theorem about existence of eigenvalues on complex vector spaces， most books must define determinants， prove that a linear map is not invertible if and only ff its determinant equals O， and then define the characteristic polynomial. This tortuous （torturous？） path gives students little feeling for why eigenvalues must exist.
　　In contrast， the simple determinant-free proofs presented here offer more insight. Once determinants have been banished to the end of the book， a new route opens to the main goal of linear algebra-understanding the structure of linear operators.
　　This book starts at the beginning of the subject， with no prerequi-sites other than the usual demand for suitable mathematical maturity.Even if your students have already seen some of the material in the first few chapters， they may be unaccustomed to working exercises of the type presented here， most of which require an understanding of proofs.
　　Vector spaces are defined in Chapter 1， and their basic propertiesare developed.

线性代数（第2版）（英文影印版） [LINEAR ALGEBRA DONE RIGHT 2nd ed] 电子书 下载 mobi epub pdf txt

评分☆☆☆☆☆

作者是美国著名数学家,用独特的方法教授 线性代数 看完使我们恍然大悟,豁然开朗.

评分☆☆☆☆☆

好书，值得推荐购买，浅显易懂。

评分☆☆☆☆☆

很好

评分☆☆☆☆☆

写的比较精简，适合初学者

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还行

评分☆☆☆☆☆

美国最受欢迎最好的线性代数教材! 各章节条理清晰,概念定义、定理证明严谨,并且阐述动机。 课后习题难度适中,主要是加深对本章内容的理解,不是教你怎么计算的。 只要仔细读完本章,习题基本都能做。此书表明了作者认为线性代数该这么讲,对于想加深理解线性代数的,此书正是你想要的。

评分☆☆☆☆☆

特别棒～

评分☆☆☆☆☆

很难得，很清楚，很好。

评分☆☆☆☆☆

过之后能够保证一辈子忘不掉这个定理的推导过程。。。

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