I am trying to prove that components in R2 are a homomorphism. I am using phi(a,b) + phi(c,d) and rewriting this as phi(a+c), phi (b+d). I believe I understand the definition of a homomorphism for addition, phi(a+b)=phi(a) + phi(b), but don't understand how to prove it. Thank you!
I think what you're missing is a definition of phi. R2 is the domain (and codomain) of this function. What you need to prove is the function is a homomorphism.
When you have a definition for phi, you can then plug in your arbitrary values (ie a and b) and prove that phi(a+b)=phi(a) + phi(b).