FF505/FY505 – Computational Science
Week 7, Spring 2013 [pdf format]

Contents:

• Vector spaces;
• dimension;
• inner product spaces;
• orthogonal sets and orthonormal bases;
• linear transformations;
• eigenvalues;
• eigenvectors;
• the characteristic polynomial;
• eigenvalue problems;
• bases of eigenvectors;
• diagonalization.
• the Spectral theorem;
• Fourier series.

MATLAB Exercises

Work in small groups at the computer to solve the following exercises.

1. Draw the graphs of some partial sums for the Fourier series of f(x)=x³−3π/2x², and compare these with the graph of f(x).
2. Draw over the (t,x) plane some partial sums for the solution of exercise 7 in Claudio’s Exercises Part 3 (assuming for example that c=1). What happens after long time?
3. MATLAB has a utility for visualizing eigenvalues. Try help eigshow. Explore this function and understand what it does.
4. Carry out the exercises in Chapter 2 of Paolo’s Notes.