here's another problem i'm working on: http://i475.photobucket.com/albums/r...de0flife/2.jpg
i've got a few adv. linear algebra problems that i need to do, but i'm not real knowledgeable on representation theory.
here is the first and easiest problem of the bunch: problem #1
also, with that i need to find the trace and norm of 2 - 3i.
any help would be greatly appreciated.
here's another problem i'm working on: http://i475.photobucket.com/albums/r...de0flife/2.jpg
ok, this problem has given me the most trouble so far. i'm not really sure where to start:
http://i475.photobucket.com/albums/r...ber3ans010.jpg
and again, i still haven't been able to solve this problem: http://i475.photobucket.com/albums/r...de0flife/2.jpg
since is a finite group, the order of is finite, i.e. for some integer now if is an eigenvalue of then for some non-zero vector
thus hence so every eigenvalue of is a root of unity and thus now is an eigenvalue of if and only if
for some vector if and only if i.e. if and only if is an eigenvalue of so if are the eigenvalues of then the
eigenvalues of are: therefore:
the reason that i haven't answered that question is that it's quite long and, well, you should do some of the questions yourself!
anyway, for part (a), you need to use the definition of to prove two things:
1) or as the notation in your problem is an automorphism of for any this proves that is well-defined.
2) for any which will prove that is a homomorphism and thus a representation of
You're right. I should be doing these questions on my own, and i've spent hours focusing on this problem alone and haven't really gotten anywhere with it. I've even done a little research on Representation theory and i can't seem to find what i need to be able to solve this question.