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From one of the premiere author teams in higher education, comes a new linear algebra textbook that fosters mathematical thinking, problem-solving abilities, and exposure to real-world applications. Without sacrificing mathematical precision, Anton and Busby focus on the aspects of linear algebra that are most likely to have practical value to the student while not compromising the intrinsic mathematical form of the subject. Throughout Contemporary Linear Algebra, students are encouraged to look at ideas and problems from multiple points of view. Contemporary Linear Algebra meets the guidelines of the Linear Algebra Curriculum Study Group (LACSG). All major concepts are introduced early and revisited in more depth later on. This spiral approach to concept development ensures that all key topics can be covered in the course. The authors believe that a working knowledge of vectors in Rn and some experience with viewing functions as vectors is the right focus for this course. Material on Axiomatic vector spaces appears towards the end so as to avoid the wall of abstraction so many students encounter. The text provides students with a strong geometric foundation upon which to build. In keeping with this goal, the text covers vectors first then proceeds to linear systems, which allows the authors to interpret parametric solutions of linear systems as geometric objects. Looking Ahead elements provides students with insight into the future role of the material currently being studied. A wide range of applications throughout gives students a sense of the broad applicability of linerar algebra. The applications include very modern topics such as global positioning and internet search procedures.
Chapter 1. Vectors.
1.1 Vectors and Matrices in Engineering and Mathematics in-Space.
1.2 Dot Product and Orthogonality.
1.3 Vector Equations of Lines and Planes.
Chapter 2. Systems of Linear Equations.
2.1 Introduction to Systems of Linear Equations.
2.2 Solving Linear Systems by Row Reduction.
2.3 Applications of Linear Systems.
Chapter 3. Matrices and Matrix Algebra.
3.1 Operations on Matrices.
3.2 Inverses; Algebraic Properties of Matrices.
3.3 Elementary Matrices; A Method for Finding A-1.
3.4 Subspaces and Linear Independence.
3.5 The Geometry of Linear Systems.
3.6 Matrices with Special Forms.
3.7 Matrix Factorizations; LU-Decomposition.
3.8 Partitioned Matrices and Parallel Processing.
Chapter 4. Determinants.
4.1 Determinants; Cofactor Expansion.
4.2 Properties of Determinants.
4.3 Cramer's Role; Formula for A-1; Applications of Determinants.
4.4 A First Look at Eigenvalues and Eigenvectors.
Chapter 5. Matrix Models.
5.1 Dynamical Systems and Markov Chains.
5.2 Leontief Input-Output Models.
5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear Systems.
5.4 The Power Method; Application to Internet Search Engines.
Chapter 6. Linear Transformations.
6.1 Matrices as Transformations.
6.2 Geometry of Linear Operators.
6.3 Kernel and Range.
6.4 Composition and Invertibility of Linear Transformations.
6.5 Computer Graphics.
Chapter 7. Dimension and Structure.
7.1 Basis and Dimension.
7.2 Properties of Bases.
7.3 The Fundamental Spaces of a Matrix.
7.4 The Dimension Theorem and Its Implications.
7.5 The Rank Theorem and Its Implications.
7.6 The Pivot Theorem and Its Implications.
7.7 The Projection Theorem and Its Implications.
7.8 Best Approximation and Least Squares.
7.9 Orthonormal Bases and the Gram-Schmidt Process.
7.10 QR-Decomposition; Householder Transformations.
7.11 Coordinates with Respects to a Basis.
Chapter 8. Diagonalization.
8.1 Matrix Representations of Linear Transformations.
8.2 Similarity and Diagonalizability.
8.3 Orthogonal Diagonalizability; Functions of a Matrix.
8.4 Quadratic Forms.
8.5 Application of Quadratic Forms to Optimization.
8.6 Singular Value Decomposition.
8.7 The Pseudoinverse.
8.8 Complex Eigenvalues and Eigenvectors.
8.9 Hermitian, Unitary, and Normal Matrices.
8.10 Systems of Differential Equations.
Chapter 9. General Vector Spaces.
9.1 Vector Space Axioms.
9.2 Inner Product Spaces; Fourier Series.
9.3 General Linear Transformations; Isomorphism.
Appendix A. How to Read Theorems.
Appendix B. Complex Numbers.
From one of the premiere author teams in higher education, comes a new linear algebra textbook that fosters mathematical thinking, problem-solving abilities, and exposure to real-world applications. Without sacrificing mathematical precision, Anton and Busby focus on the aspects of linear algebra that are most likely to have practical value to the student while not compromising the intrinsic mathematical form of the subject. Throughout Contemporary Linear Algebra, students are encouraged to look at ideas and problems from multiple points of view. Contemporary Linear Algebra meets the guidelines of the Linear Algebra Curriculum Study Group (LACSG). All major concepts are introduced early and revisited in more depth later on. This spiral approach to concept development ensures that all key topics can be covered in the course. The authors believe that a working knowledge of vectors in Rn and some experience with viewing functions as vectors is the right focus for this course. Material on Axiomatic vector spaces appears towards the end so as to avoid the wall of abstraction so many students encounter. The text provides students with a strong geometric foundation upon which to build. In keeping with this goal, the text covers vectors first then proceeds to linear systems, which allows the authors to interpret parametric solutions of linear systems as geometric objects. Looking Ahead elements provides students with insight into the future role of the material currently being studied. A wide range of applications throughout gives students a sense of the broad applicability of linerar algebra. The applications include very modern topics such as global positioning and internet search procedures.
Table of Contents
Chapter 1. Vectors.
1.1 Vectors and Matrices in Engineering and Mathematics in-Space.
1.2 Dot Product and Orthogonality.
1.3 Vector Equations of Lines and Planes.
Chapter 2. Systems of Linear Equations.
2.1 Introduction to Systems of Linear Equations.
2.2 Solving Linear Systems by Row Reduction.
2.3 Applications of Linear Systems.
Chapter 3. Matrices and Matrix Algebra.
3.1 Operations on Matrices.
3.2 Inverses; Algebraic Properties of Matrices.
3.3 Elementary Matrices; A Method for Finding A-1.
3.4 Subspaces and Linear Independence.
3.5 The Geometry of Linear Systems.
3.6 Matrices with Special Forms.
3.7 Matrix Factorizations; LU-Decomposition.
3.8 Partitioned Matrices and Parallel Processing.
Chapter 4. Determinants.
4.1 Determinants; Cofactor Expansion.
4.2 Properties of Determinants.
4.3 Cramer's Role; Formula for A-1; Applications of Determinants.
4.4 A First Look at Eigenvalues and Eigenvectors.
Chapter 5. Matrix Models.
5.1 Dynamical Systems and Markov Chains.
5.2 Leontief Input-Output Models.
5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear Systems.
5.4 The Power Method; Application to Internet Search Engines.
Chapter 6. Linear Transformations.
6.1 Matrices as Transformations.
6.2 Geometry of Linear Operators.
6.3 Kernel and Range.
6.4 Composition and Invertibility of Linear Transformations.
6.5 Computer Graphics.
Chapter 7. Dimension and Structure.
7.1 Basis and Dimension.
7.2 Properties of Bases.
7.3 The Fundamental Spaces of a Matrix.
7.4 The Dimension Theorem and Its Implications.
7.5 The Rank Theorem and Its Implications.
7.6 The Pivot Theorem and Its Implications.
7.7 The Projection Theorem and Its Implications.
7.8 Best Approximation and Least Squares.
7.9 Orthonormal Bases and the Gram-Schmidt Process.
7.10 QR-Decomposition; Householder Transformations.
7.11 Coordinates with Respects to a Basis.
Chapter 8. Diagonalization.
8.1 Matrix Representations of Linear Transformations.
8.2 Similarity and Diagonalizability.
8.3 Orthogonal Diagonalizability; Functions of a Matrix.
8.4 Quadratic Forms.
8.5 Application of Quadratic Forms to Optimization.
8.6 Singular Value Decomposition.
8.7 The Pseudoinverse.
8.8 Complex Eigenvalues and Eigenvectors.
8.9 Hermitian, Unitary, and Normal Matrices.
8.10 Systems of Differential Equations.
Chapter 9. General Vector Spaces.
9.1 Vector Space Axioms.
9.2 Inner Product Spaces; Fourier Series.
9.3 General Linear Transformations; Isomorphism.
Appendix A. How to Read Theorems.
Appendix B. Complex Numbers.