线性代数应该这样学Liner Algebra Done Right(中文) 线性代数应该这样学Liner Algebra Done Right(中文)

GTM Hungerford Algebra 英文 可搜索版本

代数圣经级教材，包含了群，环等现代抽象代数的结构与浅析。证明和编排不是特别科学，阅读需要一定基础。

APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface ix 1 Introduction 1 1.1 Basic Notation 1 1.2 Standard Problems of Numerical Linear Algebra 1 1.3 General Techniques 2 1.3.1 Matrix Factorizations 3 1.3.2 Perturbation Theory and Condition Numbers 4 1.3.3 Effects of Roundoff Error on Algorithms 5 1.3.4 Analyzing the Speed of Algorithms 5 1.3.5 Engineering Numerical Software 6 1.4 Example: Polynomial Evaluation 7 1.5 Floating Point Arithmetic 9 1.5.1 Further Details 12 1.6 Polynomial Evaluation Revisited 15 1.7 Vector and Matrix Norms 19 1.8 References and Other Topics for Chapter 1 23 1.9 Questions for Chapter 1 24 2 Linear Equation Solving 31 2.1 Introduction 31 2.2 Perturbation Theory 32 2.2.1 Relative Perturbation Theory 35 2.3 Gaussian Elimination 38 2.4 Error Analysis 44 2.4.1 The Need for Pivoting 45 2.4.2 Formal Error Analysis of Gaussian Elimination 46 2.4.3 Estimating Condition Numbers 50 2.4.4 Practical Error Bounds 54 2.5 Improving the Accuracy of a Solution 60 2.5.1 Single Precision Iterative Refinement 62 2.5.2 Equilibration 62 2.6 Blocking Algorithms for Higher Performance 63 2.6.1 Basic Linear Algebra Subroutines (BLAS) 66 2.6.2 How to Optimize Matrix Multiplication 67 2.6.3 Reorganizing Gaussian Elimination to Use Level 3 BLAS 72 2.6.4 More About Parallelism and Other Performance Issues . 75 vi Contents 2.7 2.8 2.9 Special Linear Systems 2.7.1 Real Symmetric Positive Definite Matrices 2.7.2 Symmetric Indefinite Matrices 2.7.3 Band Matrices 2.7.4 General Sparse Matrices 2.7.5 Dense Matrices Depending on Fewer Than O(n2) Pa- rameters References and Other Topics for Chapter 2 Questions for Chapter 2 76 76 79 79 83 90 93 93 3 Linear Least Squares Problems 101 3.1 Introduction 101 3.2 Matrix Factorizations That Solve the Linear Least Squares Prob- lem 105 3.2.1 Normal Equations 106 3.2.2 QR Decomposition 107 3.2.3 Singular Value Decompos

代数学第三章部分习题答案 1. 证明在环 R 内，若 1− ab 可逆，则 1− ba 可逆. 证明: a(1− ba) = a− aba = (1− ab)a a = (1− ab)−1a(1− ba) 1− ba = 1− b[(1− ab)−1a(1− ba)] = 1− b(1− ab)−1a(1− ba) (1− ba) + b(1− ab)−1a(1− ba) = 1 [1 + b(1− ab)−1a](1− ba) = 1 所以 (1− ba) 可逆，且 (1− ba)−1 = 1 + b(1− ab)−1a. 2. 设在环 R 中元素 u 有右逆，证明下列三条等价： (1)u 有多于一个的右逆； (2)u 是一个左零因子； (3)u 不是单位. 证明: (1)⇒(2): 由 u 有右逆知 u ̸= 0, 则 u1, u2 是 u 的右逆，u1 ̸= u2，则 u(u1 − u2) = uu1 − uu2 = 1− 1 = 0 而 u1 − u2 ̸= 0，故 u 是 u1 − u2 的左零因子. (2)⇒(3): 假设 u 为单位，则 u 可逆. 对 ∀u3 ∈ R, u3 ̸= 0. 于是 u3 = 1 · u3 = u−1uu3 = u−1(uu3) ̸= 0 即 uu3 ̸= 0，故 u 不是左零因子，矛盾！因此 u 不是单位. (3)⇒(1): 假设 u 只有一个右逆 u4，对 ∀r ∈ R, r ̸= u4, 均有 ur ̸= 1 = uu4 则 u(r − u4) ̸= 0. 由 r 的任意性知 u 不是左零因子. 而 u(1− u4u) = u− uu4u = u− u = 0 因此 1 − u4u = 0，即 u4u = 1，所以 u4 是 u 的左逆，于是 u 是单位，矛盾！ 所以 u 有多于一个的右逆. 1

Linear Algebra and Its Applications (Pearson 4ed 2012) By David.C Lay

Gilbert_Strang-Linear_Algebra_and_Its_Applications_4ed 课后答案

这是Joseph J. Rotman写的抽象代数电子版，非常不错哦！

将简单的解释与大量的实际示例结合起来，提供了一种创新的线性代数教学方法。 不需要先验知识，它涵盖线性代数的各个方面-向量，矩阵和最小二乘

G Strang的经典著作： Introduction to Linear Algebra。 最新第六版，英文原版，MIT课程用书，