线性代数-(第2版)_PDF下载[90MB-百度云]（美）阿克斯勒（Axler，S.）著

内容简介

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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.

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Preface to the InstructorPreface to the StudentAcknowledgmentsCHAPTER 1　Vector Spaces　　Complex Numbers　　Definition of Vector Space　　Properties of Vector Spaces　　Subspaces　　Sums and Direct Sums　　ExercisesCHAPTER 2　Finite-Dimenslonal Vector Spaces　　Span and Linear Independence　　Bases　　Dimension　　ExercisesCHAPTER 3　Linear Maps　　Definitions and Examples　　Null Spaces and Ranges　　The Matrix of a Linear Map　　Invertibility　　ExercisesCHAPTER 4　Potynomiags　　Degree　　Complex Coefficients　　Real Coefflcients　　ExercisesCHAPTER 5　Eigenvalues and Eigenvectors　　lnvariant Subspaces　　Polynomials Applied to Operators　　Upper-Triangular Matrices　　Diagonal Matrices　　Invariant Subspaces on Real Vector Spaces　　ExercisesCHAPTER 6　Inner-Product spaces　　Inner Products　　Norms　　Orthonormal Bases　　Orthogonal Projections and Minimization Problems 　　Linear Functionals and Adjoints　　ExercisesCHAPTER 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　　ExercisesCHAPTER 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 　　ExercisesCHAPTER 10　Trace and Determinant　　Change of Basis　　Trace　　Determinant of an Operator　　Determinant of a Matrix　　Volume　　ExercisesSymbol IndexIndex