Is it possible to have T u v Tu Tv for all U and V but TK

Is it possible to have T (u + v) = T_u + T_v for all U and V but T(Ku) notequalto K T_u for at least one K and u

Solution

Yes, it is possible that a linear map is preserved under vector addition but not under scalar multiplication.

One such example would be T(x,y) = (x+2,y)

Here, for any u = (x,y) and v=(w,z)

Tu+Tv = T(u+v)

But, T(cu) !=cT(u)