MATH 5101: LIST OF TOPICS:

I.   VECTOR SPACES

 Axiomatic properties
 Subspaces
 Spanning sets
 Linear independence
 Bases and coordinates
 Dimension
 Linear functionals and covectors
 Dual of a vector space
 Bilinear functionals
 Metric
 Isomorphism between vector space and its dual

 Null space, range space
 Dimension theorem, implicit function theorem for a linear system
 Classification of linear transformations
 Invertible transformations
 Existence and uniqueness of a system of equations
 Algebraic operations with linear transformations
 The representation theorem
 Change of basis, change of representation, and the transition matrix
 Invariant subspaces, commuting operators and eigenvectors

III. INNER PRODUCT SPACES

 Inner products
 Orthogonormal bases
 Gram-Schmidt orthogonalization process
 Orthogonal matrices
 Right and left inverses
 Least squares approximation, Bessel's inequality, normal equations
 The four fundamental subspaces of a matrix
 The Fredholm alternative, uniqueness=existence
 Intersection and sum of two vector space

IV.   EIGENVALUES AND EIGENVECTORS ON REAL VECTOR SPACES

 Eigenvector basis
 Diagonalizing a matrix
 Generalized eigenvectors
 Phase portrait of a system of linear differential equations
 Powers of a matrix
 Markov processes

V.   EIGENVALUES AND EIGENVECTORS ON COMPLEX VECTOR SPACES

 Adjoint of an operator
 Hermetian operators
 Spectral theorem
 Triangularization via unitary similarity transformation
 Diagonalization of normal matrices
 Positive definite matrices
 Quadratic forms and the generalized eigenvalue problem
 Extremization with linear constraints
 Rayleigh quotient
 Singular value decomposition of a rectangular matrix
 Pseudo-inverse of a rectangular matrix

      Texts:      (1)  L.W. Johnson, Riess & Arnold: Introduction to Linear Algebra (Chapter 4)
                      (2)  G. Strang: Linear Algebra and its Applications, 3rd Edition (Selected sections from Chapters 2&3, Chapter 5&6; Appendix A)
                      (3) Larson and Edwards: Elementary Linear Algebra, 3rd Edition  (Selected sections from Chapter 8)
 
 

MATH 5102: LIST OF  TOPICS:

Vibrations:

I. INFINITE DIMENSIONAL VECTOR SPACES: EXAMPLES

 Sturm-Liouville systems: regular, periodic, and singular
 Sturm-Liouville series

Signals and Vibrations:

II.  INFINITE DIMENSIONAL VECTOR SPACES: PRINCIPLES

 Inner product spaces
 Complete metric spaces
 Hilbert spaces
  Square summable series and square integrable functions
 Least squares approximation
  Projection theorem
  Generalized Fourier coefficients
 Bessel's inequality, Parceval's equality and completeness
 Unitary transformation between Hilbert spaces

III. FOURIER THEORY

 Fourier series
  Dirichelet kernel
  Fourier's theorem on a finite domain
 Sequences leading to the Dirac delta function
 Fourier transform representation
 Change of basis in Hilbert space:
 Orthonormal wave packet representation
 Wavelet  representations (if time permits)

Signals and Vibrations in Space and Time:

 Homogeneous sytems
 Adjoint systems
 Inhomogeneous systems
 The concept of a Green's function
 Solution via Green's function
 Integral equation of a linear system via its Green's function
 Classification of integral equations
 The Fredholm alternative
 Green's function and the resolvent of the operator of a system
 Eigenfunctions and eigenvalues via residue calculus
 Branches, branch cuts, and Riemann sheets
 Singularity structure of the resolvent of a system:
  Poles and branch cuts
  Effect of boundary conditions and domain size

 Partial differential equations: hyperbolic, parabolic, and elliptic
 The Helmholtz equation and its solutions in the Euclidean plane.
  Geometry of the space of solutions
  Plane waves vs cylinder waves:
  Why, and when to use them
  Sommerfeld's integral representation
  Hankel, Bessel, and Neumann waves
  Change of basis in the space of solutions: partial waves
  Displaced cylinder waves
  The cylindrical addition theorem
  Method of steepest descent and stationary phase
 Analytic behaviour of cylinder waves
 Interior (cavity) and exterior (scattering) boundary value problems
 Spherical waves: symmetric and non-symmetric
 Cauchy problem and characteristics (if time permits)

      Texts:      (1)  U.H. Gerlach: Linear Mathematics in Infinite Dimensions (Chapter 1,3,2,4,5)
                     (2)  F.W. Byron  and R.W. Fuller: Mathematics of Classical and Quantum Physics
 

    Possible follow up courses: Math 5451 Math 5756