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线性代数-(第2版)

2020-07-24 00:00:00

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

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线性代数-(第2版) 内容简介

线性代数-(第2版) 目录