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

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- Jan 19th 2008, 04:03 PMalexminLinear Algebra question I'm stuck on
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 - Jan 19th 2008, 04:06 PMThePerfectHacker
- Jan 20th 2008, 02:23 AMOpalg
Sorry, that one doesn't work. The map given by $\displaystyle f(\bold{v})= x$ is linear, and we're looking for a nonlinear map.

The simplest one I can come up with is the map given by $\displaystyle f(x,y) = \left\{\begin{array}{ll}x&\text{if $y\ne0$,}\\ 0&\text{if $y=0$.}\end{array}\right.$

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.