Table of Contents

1. Introduction to Vectors: 1.1 Vectors and linear combinations; 1.2 Lengths and dot products; 1.3 Matrices; 2. Solving Linear Equations: 2.1 Vectors and linear equations; 2.2 The idea of elimination; 2.3 Elimination using matrices; 2.4 Rules for matrix operations; 2.5 Inverse matrices; 2.6 Elimination = factorization: A = LU; 2.7 Transposes and permutations; 3. Vector Spaces and Subspaces: 3.1 Spaces of vectors; 3.2 The nullspace of A: solving Ax = 0; 3.3 The rank and the row reduced form; 3.4 The complete solution to Ax = b; 3.5 Independence, basis and dimension; 3.6 Dimensions of the four subspaces; 4. Orthogonality: 4.1 Orthogonality of the four subspaces; 4.2 Projections; 4.3 Least squares approximations; 4.4 Orthogonal bases and Gram-Schmidt; 5. Determinants: 5.1 The properties of determinants; 5.2 Permutations and cofactors; 5.3 Cramer's rule, inverses, and volumes; 6. Eigenvalues and Eigenvectors: 6.1 Introduction to eigenvalues; 6.2 Diagonalizing a matrix; 6.3 Applications to differential equations; 6.4 Symmetric matrices; 6.5 Positive definite matrices; 6.6 Similar matrices; 6.7 Singular value decomposition (SVD); 7. Linear Transformations: 7.1 The idea of a linear transformation; 7.2 The matrix of a linear transformation; 7.3 Diagonalization and the pseudoinverse; 8. Applications: 8.1 Matrices in engineering; 8.2 Graphs and networks; 8.3 Markov matrices, population, and economics; 8.4 Linear programming; 8.5 Fourier series: linear algebra for functions; 8.6 Linear algebra for statistics and probability; 8.7 Computer graphics; 9. Numerical Linear Algebra: 9.1 Gaussian elimination in practice; 9.2 Norms and condition numbers; 9.3 Iterative methods for linear algebra; 10. Complex Vectors and Matrices: 10.1 Complex numbers; 10.2 Hermitian and unitary matrices; 10.3 The fast Fourier transform; Solutions to selected exercises; Matrix factorizations; Conceptual questions for review; Glossary: a dictionary for linear algebra; Index; Teaching codes.

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Gilbert Strang's classic, inspirational textbook is suitable for first courses in linear algebra.

About the Author

Gilbert Strang received his Ph.D. from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. He is a Professor of Mathematics at MIT and an Honorary Fellow of Balliol College. Professor Strang has published eight textbooks. He received the von Neumann Medal of the US Association for Computational Mechanics, and the Henrici Prize for applied analysis. The first Su Buchin Prize from the International Congress of Industrial and Applied Mathematics, and the Haimo Prize from the Mathematical Association of America, were awarded for his contributions to teaching around the world.