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Full Description
For in-depth Linear Algebra courses that focus on applications. This text aims to teach basic methods and algorithms used in modern, real problems that are likely to be encountered by engineering and science students - and to foster understanding of why mathematical techniques work and how they can be derived from first principles. No text goes as far (and wide) in applications. The authors present applications hand in hand with theory, leading students through the reasoning that leads to the important results, and provide theorems and proofs where needed. Because no previous exposure to linear algebra is assumed, the text can be used for a motivated entry-level class as well as advanced undergraduate and beginning graduate engineering/applied math students.
Contents
Chapter 1. Linear Algebraic Systems 1.1. Solution of Linear Systems 1.2. Matrices and Vectors 1.3. Gaussian Elimination - Regular Case 1.4. Pivoting and Permutations 1.5. Matrix Inverses 1.6. Transposes and Symmetric Matrices 1.7. Practical Linear Algebra 1.8. General Linear Systems 1.9. Determinants Chapter 2. Vector Spaces and Bases 2.1. Vector Spaces 2.2. Subspaces 2.3. Span and Linear Independence 2.4. Bases 2.5. The Fundamental Matrix Subspaces 2.6. Graphs and Incidence Matrices Chapter 3. Inner Products and Norms 3.1. Inner Products 3.2. Inequalities 3.3. Norms 3.4. Positive Definite Matrices 3.5. Completing the Square 3.6. Complex Vector Spaces Chapter 4. Minimization and Least Squares Approximation 4.1. Minimization Problems 4.2. Minimization of Quadratic Functions 4.3. Least Squares and the Closest Point 4.4. Data Fitting and Interpolation Chapter 5. Orthogonality 5.1. Orthogonal Bases 5.2. The Gram-Schmidt Process 5.3. Orthogonal Matrices 5.4. Orthogonal Polynomials 5.5. Orthogonal Projections and Least Squares 5.6. Orthogonal Subspaces Chapter 6. Equilibrium 6.1. Springs and Masses 6.2. Electrical Networks 6.3. Structures Chapter 7. Linearity 7.1. Linear Functions 7.2. Linear Transformations 7.3. Affine Transformations and Isometries 7.4. Linear Systems 7.5. Adjoints Chapter 8. Eigenvalues 8.1. Simple Dynamical Systems 8.2. Eigenvalues and Eigenvectors 8.3. Eigenvector Bases and Diagonalization 8.4. Eigenvalues of Symmetric Matrices 8.5. Singular Values 8.6. Incomplete Matrices and the Jordan Canonical Form Chapter 9. Linear Dynamical Systems 9.1. Basic Solution Methods 9.2. Stability of Linear Systems 9.3. Two-Dimensional Systems 9.4. Matrix Exponentials 9.5. Dynamics of Structures 9.6. Forcing and Resonance Chapter 10. Iteration of Linear Systems 10.1. Linear Iterative Systems 10.2. Stability 10.3. Matrix Norms 10.4. Markov Processes 10.5. Iterative Solution of Linear Systems 10.6. Numerical Computation of Eigenvalues Chapter 11. Boundary Value Problems in One Dimension 11.1. Elastic Bars 11.2. Generalized Functions and the Green's Function 11.3. Adjoints and Minimum Principles 11.4. Beams and Splines 11.5. Sturm-Liouville Boundary Value Problems 11.6. Finite Elements
