Introduction to Linear Algebra, Fifth Edition

Description

Gilbert Strang's textbooks have changed the entire approach to learning linear algebra -- away from abstract vector spaces to specific examples of the four fundamental subspaces: the column space and nullspace of A and A'. This new fifth edition has become more than a textbook for the basic linear algebra course. That is its first purpose and always will be. The new chapters about applications of the SVD, probability and statistics, and Principal Component Analysis in finance and genetics, make it also a textbook for a second course, plus a resource at work. Linear algebra has become central in modern applied mathematics. This book supports the value of understanding linear algebra. Introduction to Linear Algebra, Fifth Edition includes challenge problems to complement the review problems that have been highly praised in previous editions. The basic course is followed by eight applications: differential equations in engineering, graphs and networks, statistics, Fourier methods and the FFT, linear programming, computer graphics, cryptography, Principal Component Analysis, and singular values. Audience: Thousands of teachers in colleges and universities and now high schools are using this book, which truly explains this crucial subject. This text is for readers everywhere, with support from the websites and video lectures. Every chapter begins with a summary for efficient review. Contents: Chap. 1: Introduction to Vectors; Chap. 2: Solving Linear Equations; Chap. 3: Vector Spaces and Subspaces; Chap. 4: Orthogonality; Chap. 5: Determinants; Chap. 6: Eigenvalues and Eigenvectors; Chap. 7: Singular Value Decomposition; Chap. 8: Linear Transformations; Chap. 9: Complex Vectors and Matrices; Chap. 10: Applications; Chap. 11: Numerical Linear Algebra; Chap. 12: Linear Algebra in Probability and Statistics; Matrix Factorizations; Index; Six Great Theorems. Cover Table of Contents Preface 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 and Rx = 0 3.3 The Complete Solution to Ax = b 3.4 Independence, Basis and Dimension 3.5 Dimensions of the Four Subspaces 4 Orthogonality 4.1 Orthogonality of the Four Subspaces 4.2 Projections 4.3 Least Squares Approximations 4.4 Orthonormal 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 Systems of Differential Equations 6.4 Symmetric Matrices 6.5 Positive Definite Matrices 7 The Singular Value Decomposition (SVD) 7.1 Image Processing by Linear Algebra 7.2 Bases and Matrices in the SVD 7.3 Principal Component Analysis (PCA by the SVD) 7.4 The Geometry of the SVD 8 Linear Transformations 8.1 The Idea of a Linear Transformation 8.2 The Matrix of a Linear Transformation 8.3 The Search for a Good Basis 9 Complex Vectors and Matrices 9.1 Complex Numbers 9.2 Hermitian and Unitary Matrices 9.3 The Fast Fourier Transform 10 Applications 10.1 Graphs and Networks 10.2 Matrices in Engineering 10.3 Markov Matrices, Population, and Economics 10.4 Linear Programming 10.5 Fourier Series: Linear Algebra for Functions 10.6 Computer Graphics 10.7 Linear Algebra for Cryptography 11 Numerical Linear Algebra 11.1 Gaussian Elimination in Practice 11.2 Norms and Condition Numbers 11.3 Iterative Methods and Preconditioners 12 Linear Algebra in Probability & Statistics 12.1 Mean, Variance, and Probability 12.2 Covariance Matrices and Joint Probabilities 12.3 Multivariate Gaussian and Weighted Least Squares Matrix Factorizations Index Six Great Theorems of Linear Algebra/Linear Algebra in a Nutshell