Applied Linear Algebra: The Decoupling Principle

1. The Decoupling Principle. 2. Vector Spaces and Bases. Vector Spaces. Linear Independence, Basis and Dimension. Properties and Uses of a Basis. Change of Basis. Building New Vector Spaces from Old Ones. 3. Linear Transformations and Operators. Definitions and Examples. The Matrix of a Linear Transformation. The Effect of a Change of Basis. Infinite Dimensional Vector Spaces. Kernels, Ranges, and Quotient Maps. 4. An Introduction to Eigenvalues. Definitions and Examples. Bases of Eigenvectos. Eigenvalues and the Characteristic Polynomial. The Need for Complex Eigenvalues. When is an Operator Diagonalizable? Traces, Determinants, and Tricks of the Trade. Simultaneous Diagonalization of Two Operators. Exponentials of Complex Numbers and Matrices. Power Vectors and Jordan Canonical Form. 5. Some Crucial Applications. Discrete-Time Evolution: x(n)=Ax(n-1). First-Order Continuous-Time Evolution: dx/dt=Ax. Second-Order Continuous-Time Evolution: d2x/dt2=Ax. Reducing Second-Order Problems to First-Order. Long-Time Behavior and Stability. Markov Chains and Probability Matrices. Linear Analysis near Fixed Points of Nonlinear Problems. 6. Inner Products. Real Inner Products: Definitions and Examples. Complex Inner Products. Bras, Kets, and Duality. Expansion in Orthonormal Bases: Finding Coefficients. Projections and the Gram-Schmidt Process. Orthogonal Complements and Projections onto Subspaces. Least Squares Solutions. The Spaces l2 and L2(0,1). Fourier Series on an Interval. 7. Adjoints, Hermitian Operators, and Unitary Operators. Adjoints and Transposes. Hermitian Operators. Quadratic Forms and Real Symmetric Matrices. Rotations, Orthogonal Operators, and Unitary Operators. How the Four Classes are Related. 8. The Wave Equation. Waves on the Line. Waves on the Half Line Dirichlet and Neumann Boundary Conditions. The Vibrating String. Standing Waves and Fourier Series. Periodic Boundary Conditions. Equivalence of Traveling Waves and Standing Waves. The Different Types of Fourier Series. 9. Continuous Spectra and the Dirac Delta Function. The Spectrum of a Linear Operator. The Dirac o Function. Distributions. Generalized Eigenfunction Expansions The Spectral Theorem. 10. Fourier Transforms. Existence of Fourier Transforms. Basic Properties of Fourier Transforms. Convolutions and Differential Equations. Partial Differential Equations. Bandwidth and Heisenberg's Uncertainty Principle. Fourier Transforms on the Half Line. 11. Green's Functions. Delta Functions and the Superposition Principle. Inverting Operators. The Method of Images. Initial Value Problems. Laplace's Equation on R2. Index.