Frustrations and Loss of Inspiration... And Failures

I woke up early so that I can go to UP early to review. When I was already there, I started reviewing at the library. After a few more scans and reads, I rested and met my friends. Practically, we were "enjoying" (there were 2 birthday celebrants) the times until our professor came.

EVERYTHING IS A DISASTER! I only know 2 out of 5 items. At the end, I answered 2.5/5 items. Here are my failures:

1. linear independence of vectors <=> non-singular coordinate matrix
~>I'd done the <= part. For the => part, I was trying the contrapositive. Later did I realize that I am almost done. Sadly, I don't know what to do that time.
Points: 0.5 x 7

2. how many sol's does AX=0 have if AX=B has no sol'n?
~>My answer: the nullity of A. The answer: infinitely many. My professor said, "Mapapamura ka kapag nalaman mo yung sagot". And he was right. This is actually related to the => part of 1.
Points: 0

3. find P s.t. P'AP is diagonal
~>Easy. I almost forgot the answer. If not for the hint given by my professor earlier, I will have no answer for this item.
Points: 1 x 7

4. if A^m is 0, show I-A is non-singular
~>Very Easy. Just abuse the series expansion of 1/(1-x).
Points: 1 x 7

5. if A is an orthogonal basis, then the norm is preserved on the linear transformation
~>No comment.
Points: 0 x 7

Total: 50%. FAILING
Final Grade: 67%. => 2.75. Letse. You broke my streak. T_T

Actually, before and during the exam, I was waiting for a divine intervention, but nothing came. Good thing, at home, there was popcorn and soda. At least they made me happy.