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內容簡介

　　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.

內頁插圖

目錄

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.

綫性代數（第2版）（英文影印版） [LINEAR ALGEBRA DONE RIGHT 2nd ed] 下載 mobi epub pdf txt 電子書

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《綫性代數（第2版）》為高等院校理工科教材，全書共7章，內容包括：行列式；矩陣；綫性方程組；嚮量空間與綫性變換；特徵值和特徵嚮量，矩陣的對角化；二次型及應用問題，書末附錄中還介紹瞭內積空間，埃爾米特二次型；約當（Jordan）標準形；並匯編瞭曆年碩士研究生入學考試中的綫生代數試題。本書內容豐富，層次清晰，闡述深入淺齣，簡明扼要，可作為高等院校的教材（適用於35-70課的教學）或教學參考書和考研復習用書。《綫性代數（數學專業用）》是普通高等教育“十五”國傢級規劃教材，是作者主講的國傢級精品課程“綫性代數”所使用的教材。適閤作為大學本科數學類專業綫性代數（或稱“高等代數”）課程的教材，也可作為各類大專院校師生的參考書，以及關心綫性代數和矩陣論知識的科技工作者或其他讀者的自學讀物或參考書。

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