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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
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