Applied linear algebra

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. DeterminantsChapter 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 MatricesChapter 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 SpacesChapter 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 InterpolationChapter 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 SubspacesChapter 6. Equilibrium 6.1. Springs and Masses 6.2. Electrical Networks 6.3. StructuresChapter 7. Linearity 7.1. Linear Functions 7.2. Linear Transformations 7.3. Affine Transformations and Isometries 7.4. Linear Systems 7.5. AdjointsChapter 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 FormChapter 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 ResonanceChapter 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 EigenvaluesChapter 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