線性代數(shù)

前言

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.

內(nèi)容概要

　　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.

作者簡介

作者：(美國)阿克斯勒(Sheldon Axler)

書籍目錄

Preface to the InstructorPreface to the StudentAcknowledgmentsCHAPTER 1Vector SpacesComplex NumbersDefinition of Vector SpaceProperties of Vector SpacesSubspacesSums and Direct SumsExercisesCHAPTER 2Finite-Dimenslonal Vector SpacesSpan and Linear IndependenceBasesDimensionExercisesCHAPTER 3Linear MapsDefinitions and ExamplesNull Spaces and RangesThe Matrix of a Linear MapInvertibilityExercisesCHAPTER 4PotynomiagsDegreeComplex CoefficientsReal CoefflcientsExercisesCHAPTER 5Eigenvalues and Eigenvectorslnvariant SubspacesPolynomials Applied to OperatorsUpper-Triangular MatricesDiagonal MatricesInvariant Subspaces on Real Vector SpacesExercisesCHAPTER 6Inner-Product spacesInner ProductsNormsOrthonormal BasesOrthogonal Projections and Minimization Problems Linear Functionals and AdjointsExercisesCHAPTER 7Operators on Inner-Product SpacesSelf-Adjoint and Normal OperatorsThe Spectral TheoremNormal Operators on Real Inner-Product SpacesPositive OperatorsIsometriesPolar and Singular-Value DecompositionsExercisesCHAPTER 8Operators on Complex Vector SpacesGeneralized EigenvectorsThe Characteristic PolynomialDecomposition of an OperatorSquare RootsThe Minimal PolynomialJordan Form  Exercises  CHAPTER 9Operators on Real Vector SpacesEigenvalues of Square MatricesBlock Upper-Triangular MatricesThe Characteristic Polynomial  ExercisesCHAPTER 10Trace and DeterminantChange of BasisTraceDeterminant of an OperatorDeterminant of a MatrixVolumeExercisesSymbol IndexIndex

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