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Introduction To Linear Algebra 6th Edition by Gilbert Strang, ISBN-13: 978-1733146678
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- Publisher: Wellesley-Cambridge Press; 6th edition (April 30, 2023)
- Language: English
- 440 pages
- ISBN-10: 1733146679
- ISBN-13: 978-1733146678
The sixth edition of Strang’s phenomenally popular textbook features a more active start and new chapters on optimization and learning from data.
Linear algebra now rivals or surpasses calculus in importance for people working in quantitative fields of all kinds: engineers, scientists, economists and business people. Gilbert Strang has taught linear algebra at MIT for more than 50 years and the course he developed has become a model for teaching around the world. His video lectures on MIT OpenCourseWare have been viewed over ten million times and his twelve textbooks are popular with readers worldwide. This sixth edition of Professor Strang’s most popular book, Introduction to Linear Algebra, introduces the ideas of independent columns and the rank and column space of a matrix early on for a more active start. Then the book moves directly to the classical topics of linear equations, fundamental subspaces, least squares, eigenvalues and singular values – in each case expressing the key idea as a matrix factorization. The final chapters of this edition treat optimization and learning from data: the most active application of linear algebra today. Everything is explained thoroughly in Professor Strang’s characteristic clear style. It is sure to delight and inspire the delight and inspire the next generation of learners.
Table of Contents:
1 Vectors and Matrices 1
1.1 Vectors and Linear Combinations . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Lengths and Angles from Dot Products . . . . . . . . . . . . . . . . . . . 9
1.3 Matrices and Their Column Spaces . . . . . . . . . . . . . . . . . . . . . 18
1.4 Matrix Multiplication AB and CR . . . . . . . . . . . . . . . . . . . . 27
2 Solving Linear Equations Ax = b 39
2.1 Elimination and Back Substitution . . . . . . . . . . . . . . . . . . . . . 40
2.2 Elimination Matrices and Inverse Matrices . . . . . . . . . . . . . . . . . 49
2.3 Matrix Computations and A = LU . . . . . . . . . . . . . . . . . . . . 57
2.4 Permutations and Transposes . . . . . . . . . . . . . . . . . . . . . . . . 64
3 The Four Fundamental Subspaces 75
3.1 Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Computing the Nullspace by Elimination:A=CR . . . . . . . . . . . . 84
3.3 The Complete Solution to Ax = b . . . . . . . . . . . . . . . . . . . . . 95
3.4 Independence, Basis, and Dimension . . . . . . . . . . . . . . . . . . . . 106
3.5 Dimensions of the Four Subspaces . . . . . . . . . . . . . . . . . . . . . 120
4 Orthogonality 135
4.1 Orthogonality of Vectors and Subspaces . . . . . . . . . . . . . . . . . . 136
4.2 Projections onto Lines and Subspaces . . . . . . . . . . . . . . . . . . . 143
4.3 Least Squares Approximations . . . . . . . . . . . . . . . . . . . . . . . 155
4.4 Orthonormal Bases and Gram-Schmidt . . . . . . . . . . . . . . . . . . . 168
4.5 The Pseudoinverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 182
5 Determinants 191
5.1 3 by 3 Determinants and Cofactors . . . . . . . . . . . . . . . . . . . . . 192
5.2 Computing and Using Determinants . . . . . . . . . . . . . . . . . . . . 198
5.3 Areas and Volumes by Determinants . . . . . . . . . . . . . . . . . . . . 204
6 Eigenvalues and Eigenvectors 209
6.1 Introduction to Eigenvalues : Ax = x . . . . . . . . . . . . . . . . . . 210
6.2 Diagonalizing a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.3 Symmetric Positive Definite Matrices . . . . . . . . . . . . . . . . . . . 239
6.4 Complex Numbers and Vectors and Matrices . . . . . . . . . . . . . . . 255
6.5 Solving Linear Differential Equations . . . . . . . . . . . . . . . . . . . 263
7 The Singular Value Decomposition (SVD) 286
7.1 Singular Values and Singular Vectors . . . . . . . . . . . . . . . . . . . . 287
7.2 Image Processing by Linear Algebra . . . . . . . . . . . . . . . . . . . . 297
7.3 Principal Component Analysis (PCA by the SVD) . . . . . . . . . . . . . 302
8 Linear Transformations 308
8.1 The Idea of a Linear Transformation . . . . . . . . . . . . . . . . . . . . 309
8.2 The Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . 318
8.3 The Search for a Good Basis . . . . . . . . . . . . . . . . . . . . . . . . 327
9 Linear Algebra in Optimization 335
9.1 Minimizing a Multivariable Function . . . . . . . . . . . . . . . . . . . . 336
9.2 Backpropagation and Stochastic Gradient Descent . . . . . . . . . . . . . 346
9.3 Constraints, LagrangeMultipliers, Minimum Norms . . . . . . . . . . . 355
9.4 Linear Programming, Game Theory, and Duality . . . . . . . . . . . . . 364
10 Learning from Data 370
10.1 Piecewise Linear Learning Functions . . . . . . . . . . . . . . . . . . . . 372
10.2 Creating and Experimenting . . . . . . . . . . . . . . . . . . . . . . . . 381
10.3 Mean, Variance, and Covariance . . . . . . . . . . . . . . . . . . . . . . 386
Appendix 1 The Ranks of AB and A + B 400
Appendix 2 Matrix Factorizations 401
Appendix 3 Counting Parameters in the Basic Factorizations 403
Appendix 4 Codes and Algorithms for Numerical Linear Algebra 404
Appendix 5 The Jordan Form of a Square Matrix 405
Appendix 6 Tensors 406
Appendix 7 The Condition Number of a Matrix Problem 407
Appendix 8 Markov Matrices and Perron-Frobenius 408
Appendix 9 Elimination and Factorization 410
Appendix 10 Computer Graphics 413
Index of Equations 419
Index of Notations 422
Index 423
Gilbert Strang is a professor of mathematics at the Massachusetts Institute of Technology, where his research focuses on analysis, linear algebra and PDEs. He is the author of many textbooks and his service to the mathematics community is extensive. He has spent time both as President of SIAM and as Chair of the Joint Policy Board for Mathematics, and has been a member of various other committees and boards. He has received several awards for his research and teaching, including the Chauvenet Prize (1976), the Award for Distinguished Service (SIAM, 2003), the Graduate School Teaching Award (Massachusetts Institute of Technology, 2003) and the Von Neumann Prize Medal (US Association for Computational Mechanics, 2005), among others. He is a Member of the National Academy of Sciences, a Fellow of the American Academy of Arts and Sciences, and an Honorary Fellow of Balliol College, Oxford.
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