Applied Linear Algebra

Quiz 4

Due 7 May 1999

This is a take-home group quiz. A group should consist of 3-4 students. One solution set per group is accepted.

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

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

   

   Find all its eigenvalues, eigenvectors, generalized eigenvectors, and construct its Jordan form and the minimal annihilating polynomial.

2. 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, consisting of all polynomials with real coefficients having degree at most 1. Let V be equipped with the following scalar product:

   

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

3. Let V be the vector space over the field of real numbers spanned by functions and where is an independent variable. Let V be equipped with the following scalar product:

   

   Let A be a linear mapping defined by the following:

   

   Is A symmetric? Antisymmetric?