線性代數

《線性代數》是Undergraduate Texts in Mathematics叢書之一。內容包括大學高年級學生為升入研究生的知識，如復矢量空間，復內積，正常算子用的譜定理，雙數空間，*小多項式，若而當典範形，有理典範形，*後一章介紹行列式。《線性代數》與同類書相比，別具特色。例如，高斯消元法被作為一個獲得特徵值得主要工具，各章有美妙的習題，等等。讀者對象：數學專業大學高年級本科生

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 ^[2]