线性代数-(第2版) 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.

线性代数-(第2版) 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

自然科学 数学 代数数论组合理论

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