Introduction to Linear Algebra, Fourth Edition

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'. Introduction to Linear Algebra, Fourth Edition includes challenge problems to complement the review problems that have been highly praised in previous editions. The basic course is followed by seven applications: differential equations, engineering, graph theory, statistics, fourier methods and the FFT, linear programming, and computer graphics. Thousands of teachers in colleges and universities and now high schools are using this book, which truly explains this crucial subject. Chapter 1: Introduction to Vectors; Chapter 2: Solving Linear Equations; Chapter 3: Vector Spaces and Subspaces; Chapter 4: Orthogonality; Chapter 5: Determinants; Chapter 6: Eigenvalues and Eigenvectors; Chapter 7: Linear Transformations; Chapter 8: Applications; Chapter 9: Numerical Linear Algebra; Chapter 10: Complex Vectors and Matrices; Solutions to Selected Exercises; Final Exam. Matrix Factorizations. Conceptual Questions for Review. Glossary: A Dictionary for Linear Algebra Index Teaching Codes Linear Algebra in a Nutshell. Table of Contents......Page 4 Preface......Page 6 1: Introduction to Vectors......Page 12 1.1 Vectors and Linear Combinations......Page 13 Review of the key ideas......Page 17 Problem Set 1.1......Page 19 1.2 Lengths and Dot Products......Page 22 1.3 Matrices......Page 33 2.1 Vectors and Linear Equations......Page 42 2.2 The Idea of Elimination......Page 56 2.3 Elimination Using Matrices......Page 68 2.4 Rules for Matrix Operations......Page 79 2.5 Inverse Matrices......Page 93 2.6 Elimination = Factorization: A=LU......Page 107 2.7 Transposes and Permutations......Page 119 3.1 Spaces of Vectors......Page 131 3.2 The Nullspace of A: Solving Ax = 0......Page 143 3.3 The Rank and the Row Reduced Form......Page 155 3.4 The Complete Solution to Ax = b......Page 166 3.5 Independence, Basis and Dimension......Page 179 3.6 Dimensions of the Four Subspaces......Page 195 4.1 Orthogonality of the Four Subspaces......Page 206 4.2 Projections......Page 217 4.3 Least Squares Approximations......Page 229 4.4 Orthogonal Bases and Gram-Schmidt......Page 241 5.1 The Properties of Determinants......Page 255 5.2 Permutations and Cofactors......Page 266 5.3 Cramer's Rule, Inverses, and Volumes......Page 280 6.1 Introduction to Eigenvalues......Page 294 6.2 Diagonalizing a Matrix......Page 309 6.3 Applications to Differential Equations......Page 323 6.4 Symmetric Matrices......Page 341 6.5 Positive Definite Matrices......Page 353 6.6 Similar Matrices......Page 366 6.7 Singular Value Decomposition (SVD)......Page 374 7.1 The Idea of a Linear Transformation......Page 386 7.2 The Matrix of a Linear Transformation......Page 395 7.3 Diagonalization and the Pseudoinverse......Page 410 8.1 Matrices in Engineering......Page 420 8.2 Graphs and Networks......Page 431 8.3 Markov Matrices, Population, and Economics......Page 442 8.4 Linear Programming......Page 451 8.5 Fourier Series: Linear Algebra for Functions......Page 458 8.6 Linear Algebra for Statistics and Probability......Page 464 8.7 Computer Graphics......Page 470 9.1 Gaussian Elimination in Practice......Page 476 9.2 Norms and Condition Numbers......Page 486 9.3 Iterative Methods and Preconditioners......Page 492 10.1 Complex Numbers......Page 504 10.2 Hermitian and Unitary Matrices......Page 512 10.3 The Fast Fourier Transform......Page 520 Solutions to Selected Exercises......Page 527 Conceptual Questions for Review......Page 563 Glossary: A Dictionary for Linear Algebra......Page 568 Matrix Factorizations......Page 575 Teaching Codes......Page 577 Index......Page 578 Linear Algebra in a Nutshell......Page 585 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'. __Introduction to Linear Algebra, Fourth Edition__ includes challenge problems to complement the review problems that have been highly praised in previous editions. The basic course is followed by seven applications: differential equations, engineering, graph theory, statistics, fourier methods and the FFT, linear programming, and computer graphics. Thousands of teachers in colleges and universities and now high schools are using this book, which truly explains this crucial subject. Chapter 1: Introduction to Vectors; Chapter 2: Solving Linear Equations; Chapter 3: Vector Spaces and Subspaces; Chapter 4: Orthogonality; Chapter 5: Determinants; Chapter 6: Eigenvalues and Eigenvectors; Chapter 7: Linear Transformations; Chapter 8: Applications; Chapter 9: Numerical Linear Algebra; Chapter 10: Complex Vectors and Matrices; Solutions to Selected Exercises; Final Exam. Matrix Factorizations. Conceptual Questions for Review. Glossary: A Dictionary for Linear Algebra Index Teaching Codes Linear Algebra in a Nutshell. Book 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'. Introduction To Linear Algebra, Fourth Edition Includes Challenge Problems To Complement The Review Problems That Have Been Highly Praised In Previous Editions. The Basic Course Is Followed By Seven Applications: Differential Equations, Engineering, Graph Theory, Statistics, Fourier Methods And The Fft, Linear Programming, And Computer Graphics. Thousands Of Teachers In Colleges And Universities And Now High Schools Are Using This Book, Which Truly Explains This Crucial Subject. Introduction To Vectors -- Solving Linear Equations -- Vector Spaces And Subspaces -- Orthogonality -- Determinants -- Eigenvalues And Eigenvectors -- Linear Transformations -- Applications -- Numerical Linear Algebra -- Complex Vectors And Matrices. Gilbert Strang. Includes Index. This leading textbook for first courses in linear algebra comes from the hugely experienced MIT lecturer and author Gilbert Strang. The book's tried and tested approach is direct, offering practical explanations and examples, while showing the beauty and variety of the subject. Unlike most other linear algebra textbooks, the approach is not a repetitive drill. Instead it inspires an understanding of real mathematics. The book moves gradually and naturally from numbers to vectors to the four fundamental subspaces. This new edition includes challenge problems at the end of each section. Preview five complete sections at math.mit.edu/linearalgebra. Readers can also view freely available online videos of Gilbert Strang's 18.06 linear algebra course at MIT, via OpenCourseWare (ocw.mit.edu), that have been watched by over a million viewers. Also on the web ( readers will find years of MIT exam questions, MATLAB help files and problem sets to practise what they have learned. This Informally Written Text Provides Students With A Clear Introduction Into The Subject Of Linear Algebra. Topics Covered Include Matrix Multiplication, Row Reduction, Matrix Inverse, Orthogonality And Computation. The Self-teaching Book Is Loaded With Examples And Graphics And Provides A Wide Array Of Probing Problems, Accompanying Solutions, And A Glossary.