Elementary Linear Algebra, 11th Edition
November 2013, ©2014
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
3.4 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
4.3 Linear Independence
4.4 Coordinates and Basis
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
5.3 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
8.4 Matrices for General Linear Transformations
C H A P T E R 9 Numerical Methods
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
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
- New Exercises: Hundreds of new exercises of all types have been added throughout the text.
- Technology Exercises requiring technology such as MATLAB, Mathematica, or Maple have been added and supporting data sets have been posted on the companion websites for this text. The use of technology is not essential, and these exercises can be omitted without affecting the flow of the text.
- Early Introduction to Linear Transformations: Parts of Chapter 4 and Chapter 8 were reorganized to allow for the earlier introduction to linear transformations.
- Appendix A Rewritten: The appendix on reading and writing proofs has been expanded to better support courses that focus on proving theorems.
- New Appendix C: This new appendix has been added to provide guidance on using technology software with the text.
- Web Materials: Supplementary web materials now include various applications modules, three modules on linear programming, and an alternative presentation of determinants based on permutations.
- Highlights Relationships among Concepts – By continually revisiting the web of relationships among systems of equations, matrices, determinants, vectors, linear transformations, and eigenvalues, Anton helps students to perceive linear algebra as a cohesive subject rather than as a collection of isolated definitions and techniques.
- Proof Sketches – Students sharpen their mathematical reasoning skills and understanding of proofs by filling in justifications for proof steps in some exercises.
- Emphasizes Visualization – Geometric aspects of various topics are emphasized, to support visual learners, and to provide an additional layer of understanding for all students. The geometric approach naturally leads to contemporary applications of linear algebra in computer graphics that are covered in the text.
- Mathematically Sound – Mathematical precision appropriate for mathematics majors is maintained in a book whose explanations and pedagogy meet the needs of engineering, science, and business/economics students.
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