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Elementary Linear Algebra


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Table of Contents

C H A P T E R 1 Systems of Linear Equations and Matrices 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses; Algebraic Properties of Matrices 1.5 Elementary Matrices and a Method for Finding A?1 1.6 More on Linear Systems and Invertible Matrices 1.7 Diagonal, Triangular, and Symmetric Matrices 1.8 Matrix Transformations 1.9 Applications of Linear Systems ? Network Analysis (Traffic Flow) ? Electrical Circuits ? Balancing Chemical Equations ? Polynomial Interpolation 1.10 Application: Leontief Input-Output Models C H A P T E R 2 Determinants 2.1 Determinants by Cofactor Expansion 2.2 Evaluating Determinants by Row Reduction 2.3 Properties of Determinants; Cramer?s Rule C H A P T E R 3 Euclidean Vector Spaces 3.1 Vectors in 2-Space, 3-Space, and n-Space 3.2 Norm, Dot Product, and Distance in Rn 3.3 Orthogonality 3.4 The Geometry of Linear Systems 3.5 Cross Product C H A P T E R 4 General Vector Spaces 4.1 Real Vector Spaces 4.2 Subspaces 4.3 Linear Independence 4.4 Coordinates and Basis 4.5 Dimension 4.6 Change of Basis 4.7 Row Space, Column Space, and Null Space 4.8 Rank, Nullity, and the Fundamental Matrix Spaces 4.9 Basic Matrix Transformations in R2 and R3 4.10 Properties of Matrix Transformations 4.11 Application: Geometry of Matrix Operators on R2 C H A P T E R 5 Eigenvalues and Eigenvectors 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 5.4 Application: Differential Equations 5.5 Application: Dynamical Systems and Markov Chains C H A P T E R 6 Inner Product Spaces 6.1 Inner Products 6.2 Angle and Orthogonality in Inner Product Spaces 6.3 Gram?Schmidt Process; QR-Decomposition 6.4 Best Approximation; Least Squares 6.5 Application: Mathematical Modeling Using Least Squares 6.6 Application: Function Approximation; Fourier Series C H A P T E R 7 Diagonalization and Quadratic Forms 7.1 Orthogonal Matrices 7.2 Orthogonal Diagonalization 7.3 Quadratic Forms 7.4 Optimization Using Quadratic Forms 7.5 Hermitian, Unitary, and Normal Matrices C H A P T E R 8 General Linear Transformations 8.1 General Linear Transformation 8.2 Compositions and Inverse Transformations 8.3 Isomorphism 8.4 Matrices for General Linear Transformations 8.5 Similarity C H A P T E R 9 Numerical Methods 9.1 LU-Decompositions 9.2 The Power Method 9.3 Comparison of Procedures for Solving Linear Systems 9.4 Singular Value Decomposition 9.5 Application: Data Compression Using Singular Value Decomposition C H A PT E R 10 Applications of Linear Algebra 10.1 Constructing Curves and Surfaces Through Specified Points 10.2 The Earliest Applications of Linear Algebra 10.3 Cubic Spline Interpolation 10.4 Markov Chains 10.5 Graph Theory 10.6 Games of Strategy 10.7 Leontief Economic Models 10.8 Forest Management 10.9 Computer Graphics 10.10 Equilibrium Temperature Distributions 10.11 Computed Tomography 10.12 Fractals 10.13 Chaos 10.14 Cryptography 10.15 Genetics 10.16 Age-Specific Population Growth 10.17 Harvesting of Animal Populations 10.18 A Least Squares Model for Human Hearing 10.19 Warps and Morphs 10.20 Internet Search Engines A P P E N D I X A Working with Proofs A P P E N D I X B Complex Numbers Answers to Exercises Index

About the Author

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.

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