Applied Linear Algebra
by Lorenzo Sadun

Table of Contents

1.    The Decoupling Principle

2.    Vector Spaces and Bases
    2.1  Vector spaces
    2.2  Linear independence, basis and dimension
    2.3  Properties and uses of a basis
    2.4  Change of basis
    2.5  Building new vector spaces from old ones

3.    Linear Transformations and Operators
    3.1 Definitions and examples
    3.2 The matrix of a linear transformation
    3.3 The effect of a change of basis
    3.4  Infinite dimensional vector spaces
    3.5  Kernel, ranges and quotient maps

4.    An introduction to Eigenvalues
    4.1  Definitions and examples
    4.2  Bases of eigenvectors
    4.3  Eigenvalues and the characteristic polynomial
    4.4  The need for complex eigenvalues
    4.5  When is an operator diagonalizable?
    4.6  Traces, determinants and tricks of the trade
    4.7  Simultaneous diagonalization of two operators
    4.8  Exponentials of complex numbers and matrices
    4.9  Power vectors and Jordan canonical form

5.    Some Crucial Applications
    5.1  Discrete-time evolution: x(n)= A x(n-1)
    5.2  First-order continuous-time evolution
    5.3  Second-order continuous time evolution
    5.4  Reducing second-order problems to first order
    5.5  Long-time behavior and stability
    5.6  Markov chains and probability matrices
    5.7  Linear analysis near fixed points

6.    Inner Products
    6.1  Real inner products
    6.2  Complex inner products
    6.3  Bras, kets and duality
    6.4  Expansion in orthonormal bases
    6.5  Projections and the Gram-Schmidt process
    6.6  Orthogonal complements and projections onto subspaces
    6.7  Least squares solutions
    6.8  The spaces ell_2 and L^2[0,1]
    6.9  Fourier series on an interval

7.    Adjoints, Hermitian Operators and Unitary Operators
    7.1  Adjoints and transposes
    7.2  Hermitian operators
    7.3  Quadratic forms and real symmetric matrices
    7.4  Rotations, orthogonal operators and unitary operators
    7.5  How the four classes are related

8.    The Wave Equation
    8.1  Waves on the line
    8.2  Waves on the half line; Dirichlet and Neumann boundary conditions
    8.3  The vibrating string
    8.4  Standing waves and Fourier series
    8.5  Periodic boundary conditions
    8.6  Equivalence of traveling waves and standing waves
    8.7  The differenty types of Fourier series

9.    Continuous Spectra and the Dirac Delta Function
    9.1  The spectrum of a linear operator
    9.2  The Dirac delta function
    9.3  Distributions
    9.4  Generalized eigenfunctions; the spectral theorem

10.    Fourier Transforms
    10.1 Existence of Fourier transforms
    10.2  Basic properties
    10.3  Convolutions and differential equations
    10.4  Partial differential equations
    10.5  Bandwidth and Heisenberg's Uncertainty Principle
    10.6  Fourier transforms on the half line

11.    Green's Functions
    11.1  Delta functions and the superposition principle
    11.2  Inverting operators
    11.3  Inverting operators
    11.4  Initial value problems
    11.5  Laplace's equation on R^2