Problem 2. Take vectors v1, v2, ..., vk in Rn. Compute the projection of a vector u onto the orthogonal complement of the v1, v2, ..., vk.
Problem 3. Take the time series of the returns of the S&P 500 components over the last two years. Compute the covariance matrix of these return streams, and plot the eigenvalues of the matrix. What can you say about the "true" dimension of the space? Now, instead of returns, take the series of exponential moving averages of the returns with span=60. How does the covariance matrix and its eigenvalues change?
Problem 4. With the data as above, look at the distribution of the entries of the correlation (not covariance) matrix. Plot its histogram. Look at highest correlations, and try to figure out if this is due to actual relationships between the stocks or is purely random (to check on the latter, see what happens over the preceding two year period).