STAT 5645/6645 - Fall semester 2024
Instructions: Due Date: Wed. Oct. 23 11:59PM.
This exam is open materials. Here are the rules for the two course levels:
- 5645 Students: You may discuss problems and solution strategies together, but your submitted work must be your own. You may also seek help from Haozhe or Dr. Walsh
- 6645 Students: You must work individually on this exam, but you may discuss problems and solutions with Haozhe or Dr. Walsh.
Write your solutions as well-organized narratives or coherent mathematical arguments with complex steps justified via invoking a theorem, definition, or other results. If you don't quite understand a problem when you hand in the exam, show your work and what you were thinking on the problem - I will award partial credit for hard work that moves in the direction of a solution.
You can use any computer program to help solve a problem as long as you explain what calculations you used, how, and why as part of your solution.
Honesty Statement: I have followed the instructions for this exam. The work here represents my own. I have not cheated nor done anything like cheating on this exam.
Matrix Algebra and Vector Space Concepts
1. (15pts - 5645) (5pts - 6645) Problem 2.11 in the book.
2. (15pts - 5645) (5pts - 6645) Let matrices A and B be conformable for both AB and BA multiplications.
(a) Show that \( \operatorname{tr}(\mathbf{A B})=\operatorname{tr}(\mathbf{B A}) \).
(b) Extend the result to \( \operatorname{tr}(\mathbf{A B C})=\operatorname{tr}(\mathbf{C A B})=\operatorname{tr}(\mathbf{B C A}) \).
3. (20pts - 5645) (10pts - 6645) Consider the linear transformation given by the matrix \( A \) with respect to the standard basis.
