Courses Detail Information
MATH6001J – Methods of Applied Mathematics I
Instructors:
Credits: 3 Credits
Pre-requisites: Graduate standing or permission of instructor.
Description:
In the first part, notions of linear algebra are reviewed and extended to infinite-dimensional vector spaces. This includes concepts such as scalar products, norms and (Schauder-) bases. As an application, Legendre polynomials, introduced as an orthonormalization of the monomials on the interval [-1,1] are introduced, and their role in multipole expansions is explored. Next, Hilbert spaces are introduced, leading to spaces of square-integrable functions and Fourier series. A look back and comparison of the obtained results with the finite-dimensional cases of linear algebra concludes this part.
The second part focuses on bounded linear maps on (infinite-dimensional) spaces, introducing the matrix elements of such operators and using these to define Hilbert-Schmidt operators for square-summable sequences and square-integrable functions. The notions of inverses and adjoints of bounded linear operators are discussed and the spectrum of such operators is introduced. Compact operators are introduced and, motivated by a question from the theory of partial differential equations, the spectral theorem for compact operators is established.
The last part is dedicated to applications of the spectral theory, including the Rayleigh-Ritz method (applied specifically to Sturm-Liouville eigenvalue problems) and the polar and singular value decompositions of compact operators, which of course includes these decompositions for matrices.
Course Topics:
Normed Vector Spaces
Bases and Inner Product Spaces
Legendre Polynomials and Applications
Hilbert Spaces
The Space of Square-Integrable Functions
Fourier Series
Finite-Dimensional Vector Spaces
Linear Functionals and Operators
Matrix Elements and Hilbert-Schmidt Operators
Inverse and Adjoint of Bounded Linear Operators
The Spectrum
Compact Operators
Spectral Theorem for Compact Operators
Sturm-Liouville Boundary Value Problems
The Rayleigh-Ritz Method
Positive Operators and the Polar Decomposition
The Singular Value Decomposition for Compact Operators