Linear maps satisfy both conditions L1 and L2 below. I have to find a map R^2 -> R that satisfies ONLY L2 and not L1. I have been thinking about this for quite a while, but I haven't made that much progress. Do I have to use absolute values in some way?
It would be nice to see the actual solution or at least get some clear hints.
x and y are vectors.
L1: f(x+y) = f(x) + f(y)
L2: af(x) = f (ax)
Your polar coordinate hint led me to try trigonometric functions, but I couldn't find a solution there. I have been trying pretty much every possible polar coordinate function I can think of, but the problem is that I just can't seem to find this "suitable" function. Whenever I find a map that satisfies L2, it always satisfies L1 too. More help would be apprectiated.