Contemporary Linear Algebra

Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his Ph.D. from the Polytechnic Institute of Brooklyn, all in mathematics. He worked in the manned space program at Cape Canaveral in the early 1960's. In 1968 he became a research professor of mathematics at Drexel University in Philadelphia, where he taught and did mathematical research for 15 years. In 1983 he left Drexel as a Professor Emeritus of Mathematics to become a full-time writer of mathematical textbooks. There are now more than 150 versions of his books in print, including translations into Spanish, Arabic, Portuguese, French, German, Chinese, Japanese, Hebrew, Italian, and Indonesian. He was awarded a Textbook Excellence Award in 1994 by the Textbook Authors Association, and in 2011 that organization awarded his Elementary Linear Algebra text its McGuffey Award.

"Enthusiasts will find the many historical remarks of interest." (Mathematika, No.50, 2005) "...It deserves to become a popular textbook with instructor and student alike". (Zentralblatt MATH, Vol.1008, No.8, 2003)

• CHAPTER 1 Vectors 11.1 Vectors and Matrices in Engineering and Mathematics; n-Space 11.2 Dot Product and Orthogonality 151.3 Vector Equations of Lines and Planes 29CHAPTER 2 Systems of Linear Equations 392.1 Introduction to Systems of Linear Equations 392.2 Solving Linear Systems by Row Reduction 482.3 Applications of Linear Systems 63CHAPTER 3 Matrices and Matrix Algebra 793.1 Operations on Matrices 793.2 Inverses; Algebraic Properties of Matrices 943.3 Elementary Matrices; A Method for Finding A−1 1093.4 Subspaces and Linear Independence 1233.5 The Geometry of Linear Systems 1353.6 Matrices with Special Forms 1433.7 Matrix Factorizations; LU-Decomposition 1543.8 Partitioned Matrices and Parallel Processing 166CHAPTER 4 Determinants 1754.1 Determinants; Cofactor Expansion 1754.2 Properties of Determinants 1844.3 Cramer's Rule; Formula for A −1; Applications of Determinants 1964.4 A First Look at Eigenvalues and Eigenvectors 210CHAPTER 5 Matrix Models 2255.1 Dynamical Systems and Markov Chains 2255.2 Leontief Input-Output Models 2355.3 Gauss–Seidel and Jacobi Iteration; Sparse Linear Systems 2415.4 The Power Method; Application to Internet Search Engines 249CHAPTER 6 Linear Transformations 2656.1 Matrices as Transformations 2656.2 Geometry of Linear Operators 2806.3 Kernel and Range 2966.4 Composition and Invertibility of Linear Transformations 3056.5 Computer Graphics 318CHAPTER 7 Dimension and Structure 3297.1 Basis and Dimension 3297.2 Properties of Bases 3357.3 The Fundamental Spaces of a Matrix 3427.4 The Dimension Theorem and Its Implications 3527.5 The Rank Theorem and Its Implications 3607.6 The Pivot Theorem and Its Implications 3707.7 The Projection Theorem and Its Implications 3797.8 Best Approximation and Least Squares 3937.9 Orthonormal Bases and the Gram–Schmidt Process 4067.10 QR-Decomposition; Householder Transformations 4177.11 Coordinates with Respect to a Basis 428CHAPTER 8 Diagonalization 4438.1 Matrix Representations of Linear Transformations 4438.2 Similarity and Diagonalizability 4568.3 Orthogonal Diagonalizability; Functions of a Matrix 4688.4 Quadratic Forms 4818.5 Application of Quadratic Forms to Optimization 4958.6 Singular Value Decomposition 5028.7 The Pseudoinverse 5188.8 Complex Eigenvalues and Eigenvectors 5258.9 Hermitian, Unitary, and Normal Matrices 5358.10 Systems of Differential Equations 542CHAPTER 9 General Vector Spaces 5559.1 Vector Space Axioms 5559.2 Inner Product Spaces; Fourier Series 5699.3 General Linear Transformations; Isomorphism 582APPENDIX A How to Read Theorems A1APPENDIX B Complex Numbers A3ANSWERS TO ODD-NUMBERED EXERCISES A9PHOTO CREDITS C1INDEX I-1