I'm a bit confused on how to prove if a transformation is linear or not.

I know that there are 2 rules:

T(u+v) = T(u)+T(v)

and

T(ku)= kT(u)

However, I'm still not sure though how to write out the problems out in their form...Can someone explain to me step by step or give me an example?

I'm trying to learn so i can figure out the following problem. Also i tried attempting these questions and i ruled out that A is not linear. In addition, i'm not sure.. but i think C,D,E are linear. Can someone check my answers?

http://img8.imageshack.us/img8/2470/mathprobl.jpg
Thanks Alot!