Question

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

Answer 1

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

Answer 2

you want to try and prove that

T((u,v)+(w,z))=T(u,v)+T(w,z)

Answer 3

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

Answer 4

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

Answer 5

you want

\[T(\alpha+\beta)=T(\alpha)+T(\beta)\]

where

\[\alpha,\beta \in \mathbb{R}^2\]

Answer 6

\[T((u,v)+(w,z))=T(u+w,v+z)=(2(u+w)+v+z,u+w-(v+z))\]



\[T(u,v)+T(w,z)=(2u+v,u-v)+(2w+z,w-z)=(2u+v+2w+z,u-v+w-z)\]

\[=(2(u+w)+v+z,u+w-(v+z))\]

notice that they are the same. So additivity holds

Answer 7

Alright thanks for your help I get it now