How do you check that T(x,y)=(2x+y,x-y) has additivity or not?

Question

anonymous · Answer

I know that it equals T(x,y)=T(x)+T(y) but for some reason I'm having a hard time proving it

zarkon · Answer

you want to try and prove that 

T((u,v)+(w,z))=T(u,v)+T(w,z)

zarkon · Answer

T(x,y)=T(x)+T(y)  really is nonsense.

anonymous · Answer

Well then how would I go about doing that? In the book it uses x and y as a reference that's why I got kind of confused

zarkon · Answer

you want 

$$T(\alpha+\beta)=T(\alpha)+T(\beta)$$

where 

$$\alpha,\beta \in \mathbb{R}^2$$

zarkon · Answer

$$T((u,v)+(w,z))=T(u+w,v+z)=(2(u+w)+v+z,u+w-(v+z))$$



$$T(u,v)+T(w,z)=(2u+v,u-v)+(2w+z,w-z)=(2u+v+2w+z,u-v+w-z)$$

$$=(2(u+w)+v+z,u+w-(v+z))$$

notice that they are the same.  So additivity  holds

anonymous · Answer

Alright thanks for your help I get it now