Applied Linear Algebra

Quiz 3

Due ? April 1999

This is a take-home group quiz. A group should consist of no more than four 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. Let V be the vector space over the field of complex numbers spanned by functions and where is an independent variable. Let A be a linear mapping defined by the following:

   

   Find all eigenvalues and corresponding eigenvectors of A.

2. Let V be the vector space over the field of complex numbers of all polynomials with complex coefficients of one complex variable x having degree at most 2. Let A be a linear mapping defined by the following:

   

   Find all eigenvalues of A.

3. Let V be the vector space over the field of complex numbers of all polynomials with complex coefficients of one complex variable x having degree at most 2. Let

   

   be a basis of V and

   

   be another basis of V. Write down the matrix of the transformation from the basis to the basis . Find the rank of the matrix.

4. Let V be the vector space over the field of real numbers spanned by functions and where is an independent variable. Let be a given nonzero linear form. Find a basis in V such that the relation

   

   holds for every vector written as

   

   Hint: Modify the solution of Problem 5, p. 132 from the text.