Applied Linear Algebra

Final Test

Spring 1999

Cite all major theorems and show all relevant work. Please do each question on a separate page.

1. (Two points). Let A be the following matrix

   

   Find all its eigenvalues, eigenvectors, generalized eigenvectors. Construct its Jordan form. Find its characteristic and minimal polynomials.

2. Let be the characteristic polynomial and be the minimal polynomial of a matrix A. Does that give you enough information to find a Jordan form of A? If so, find it. If not, explain.

3. Let V be the vector space over the field of real numbers of real-valued continuous functions of one real variable and let V' be its subspace, Let V be equipped with the following scalar product:

   

   Find the orthogonal projection of the function onto the subspace V'.