Final Exam

The final exam is in the directory:

https://github.com/williamdemeo/Math700Homework/tree/master/tex

The due date is Monday, May 5.

Please post any questions you have below.

---

Final Exam Questions and Hints

Problem 4

Question: I don't know how to get started on Part (a). Can you give me a hint?

Answer: One possible solution to Part (a) would follow these steps:

1. Recall (or, if you're ambitious, prove) that a matrix is diagonalizable if and only if its minimal polynomial is completely reducible and has no repeated roots.

2. Let R denote the restriction of A to the invariant subspace W. Prove that the minimal polynomial of R divides the minimal polynomial of A.

3. Use 1 and 2 to draw the correct conclusion.

---

Problem 6

Question: I don't know how to get started on Part (a). Can you give me a hint?

Answer: That U is a subset of the intersection of the kernels is easy, so you should check that first. Now suppose x belongs to the intersection of the kernels. You must show that x belongs to U. Follow the hint and decompose V into a direct sum, so x = u + u', with u in U and u' in U'. The crucial step in the proof is to show u' is 0. I will leave you to figure out how to show this, with one final hint: consider an endomorphism in ann(U) that you could use, along with the equation x = u + u', to show that u' must be 0.