Gilbert Strang的《线性代数（第5版）》是一本经典线性代数教材。此书深入浅出地展示了线性代数的所有核心概念，讲述过程中恰当穿插了各种应用，体现了线性代数特别有用的思想。

"线性代数内容包括行列式、矩阵、线性方程组与向量、矩阵的特征值与特征向量、二次型及Mathematica 软件的应用等。 每章都配有习题，书后给出了习题答案。本书在编写中力求重点突出、由浅入深、 通俗易懂，努力体现教学的适用性。本书可作为高等院校工科专业的学生的教材，也可作为其他非数学类本科专业学生的教材或教学参考书。第一章：向量简介。围绕向量和点积的概念，在平面和空间中引入了线性组合和线性无关的概念。
第二章：求解线性方程组。从这个基本点出发，自然引入矩阵，高斯消元，初等矩阵，可逆矩阵等重要概念，并讲述了LU分解。
第三章：线性空间与子空间。从几何的角度来理解线性方程组，引入矩阵的秩，空间的维数等重要概念。导出线性代数基本定理。
第四章：正交。给出四个基本子空间的正交关系，引入最小二乘法，以及Gram-Schmidt正交化。
第五章：行列式。从体积的角度引入行列式，证明其各种基本性质
第六章：特征值与特征向量。从如何计算方阵的高次幂出发，给出引入二者的动机。然后讲解矩阵的对角化，对称矩阵，正定矩阵。
第七章：奇异值分解。介绍了奇异值分解这个基本定理，并给出了很多应用，例如求解常微分方程，图像压缩等。
第八章：线性变换。引入抽象的线性变换的概念，讲述线性变换的矩阵表示，对角化与伪逆。
第九章：复向量与复矩阵。讨论如何自然的引入和考虑复矩阵。然后讲解Hermitian矩阵和酉矩阵，并重点介绍了快速Fourier变换这一工程上特别有用的理论，
第十章：应用。这一章集中讲授了线性代数在各个领域中的应用。
第十一章：数值线性代数。从计算实现的角度来重新看线性代数。

Table of Contents


1 Vectors and Matrices\t1
1.1 Vectors and Linear Combinations\t2
1.2 Lengths and Angles from Dot Products\t9
1.3 Matrices and Their Column Spaces\t18
1.4 Matrix Multiplication AB and CR\t27
2 Solving Linear Equations Ax = b\t39
2.1 Elimination and Back Substitution\t40
2.2 Elimination Matrices and Inverse Matrices\t49
2.3 Matrix Computations and A = LU\t57
2.4 Permutations and Transposes\t64
2.5 Derivatives and Finite Difference Matrices\t74
3 The Four Fundamental Subspaces\t84
3.1 Vector Spaces and Subspaces\t85
3.2 Computing the Nullspace by Elimination: A = CR\t93
3.3 The Complete Solution to Ax = b\t104
3.4 Independence, Basis, and Dimension\t115
3.5 Dimensions of the Four Subspaces\t129
4 Orthogonality\t143
4.1 Orthogonality of Vectors and Subspaces\t144
4.2 Projections onto Lines and Subspaces\t151
4.3 Least Squares Approximations\t163
4.4 Orthonormal Bases and Gram-Schmidt\t176
4.5 The Pseudoinverse of a Matrix\t190
5 Determinants\t198
5.1 3 by 3 Determinants and Cofactors\t199
5.2 Computing and Using Determinants\t205
5.3 Areas and Volumes by Determinants\t211
6 Eigenvalues and Eigenvectors\t216
6.1 Introduction to Eigenvalues : Ax = λx\t217
6.2 Diagonalizing a Matrix\t232
6.3 Symmetric Positive De?nite Matrices\t246
6.4 Complex Numbers and Vectors and Matrices\t262
6.5 Solving Linear Differential Equations\t270
vii

viii\tTable of Contents
7 The Singular Value Decomposition (SVD) \t 286
7.1 Singular Values and Singular Vectors\t287
7.2 Image Processing by Linear Algebra\t297
7.3 Principal Component Analysis (PCA by the SVD)\t302
8 Linear Transformations \t 308
8.1 The Idea of a Linear Transformation\t309
8.2 The Matrix of a Linear Transformation\t318
8.3 The Search for a Good Basis\t327
9 Linear Algebra in Optimization \t 335
9.1 Minimizing a Multivariable Function\t336
9.2 Backpropagation and Stochastic Gradient Descent\t346
9.3 Constraints, Lagrange Multipliers, Minimum Norms\t355
9.4 Linear Programming, Game Theory, and Duality\t364
10 Learning from Data \t 370
10.1 Piecewise Linear Learning Functions\t372
10.2 Creating and Experimenting\t381
10.3 Mean, Variance, and Covariance\t386

Appendix 1 The Ranks of AB and A + B\t400
Appendix 2 Matrix Factorizations\t401
Appendix 3 Counting Parameters in the Basic Factorizations\t403
Appendix 4 Codes and Algorithms for Numerical Linear Algebra\t404
Appendix 5 The Jordan Form of a Square Matrix\t405
Appendix 6 Tensors\t406
Appendix 7 The Condition Number of a Matrix Problem\t407
Appendix 8 Markov Matrices and Perron-Frobenius\t408
Appendix 9 Elimination and Factorization\t410
Appendix 10 Computer Graphics\t414
Index of Equations \t 419
Index of Notations\t422
Index\t423