Give an example of a function from R^2→R such that
f(av)=af(v)
for all a in R and all v in R^2 but f is not linear
Sorry, that one doesn't work. The map given by is linear, and we're looking for a nonlinear map.
The simplest one I can come up with is the map given by
This preserves scalar multiplication: if y≠0 then f(ax,ay)=ax=af(x,y), and if y=0 then f(ax,ay)=f(x,y)=0.
It is not additive, however, because for example f(1,1) + f(1,–1) = 1+1=2, but (1,1)+(1,–1)=(2,0) and f(2,0) = 0.