線性代數(第2版)（簡體書）

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.本書是線性代數後繼課程教材,在講述上採用一種新的方法，即拋棄傳統的線性代數課本中將行列式用作貫穿始終的理論前提的模式，取而代之的是將其內容放在全書內容的最後，而將關注的重點轉移到線性代數的核心問題，即對向量空間中線性算子結構的理解上。
作者非常注意啟發性地提出概念並簡潔地證明結論。例如，在本書中給出了：不利用行列式證明一個有限維複綫性空間（或一個奇數維實向量空間）中的任何線性算子都有一個本征值的方法。在各章結尾處都配有一些很有意思的題目，以幫助讀者理解並掌握內容的重點。
本書一開始討論了向量空間，線性無關，生成空間，基以及維度。讀者在全書的前多半部分（即第七章之前）會學習到內積空間的內容，在後少半部分則會學習到有限維的譜定理。
第二版在第一版的基礎上，增加了“正交投影”與“最小化問題”的內容，同時將“自共軛算子”，“正規算子”以及“譜理論”的內容進行了重新改寫。此外，第二版還增加了新的例題與習題，並將若干定理的證明進行了簡化，一些在第一版中出現的小錯誤也已得到修正。
讀者對象：數學專業高年級的本科生、研究生以及相關專業的科研人員。
目次: 向量空間; 有限維向量空間; 線性映射; 多項式; 特徵值與特徵向量; 內積空間; 內積空間的算子; 複向量空間上的算子; 實向量空間上的算子; 跡與行列式.

作者簡介：
Sheldon Axler 1975年于加州大學伯克利分校獲得博士學位。現任舊金山州立大學科學與工程學院院長，數學系教授。1976～2004，2006～2009期間享有美國國家自然科學基金資助。除本書外，還著有：《Precalculus: A Prelude to Calculus》,《Harmonic Function Theory》，《Holomorphic Spaces》等書。

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