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