Applied Linear Algebra

Midterm Test

Spring 1999

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

1. Compute the determinant of the following n-by-n real matrix when n=10:

   

   Hint: Use the linear property of determinants, and persevere.

2. Let V be a complex vector space equipped with a basis . Let A be a linear mapping that has a right inverse. Prove that a system of vectors

   

   is a basis in V, or find a counterexample. Hint: What can you say about the null-space of A?

3. Let V be the vector space over the field of real numbers spanned by functions and where is an independent variable. Let A be a linear mapping defined by the following:

   

   Does the inverse mapping exist?

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

   

   Let L=N(A) be the null-space of the mapping. Find dimension of the factor-space V/L.