Courses Detail Information
VV556 – Methods of Applied Mathematics I
Credits: 3 Credits
Pre-requisites: Vv256 Applied Calculus IV, or Vv286 Honors Mathematics IV, or consent of instructor.
Terms Offered: Fall
This course gives an introduction to the theory of bounded linear maps on finite- and infinite-dimensional spaces.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 intro- duced, leading to spaces of square-integrable functions and Fourier series. A look back and comaprison 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 opera- tors 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 oper- ators 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 de- compositions of compact operators, which of course includes these decompositions for matrices.