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$f: V \to W $ over $K$ with $a,b \in V$ and $k \in K$.

Additivity: $f(a+b) = f(a) + f(b)$

Homogenity: $k*f(b) =f(k * b)$

I have a visual understanding that a function is linear if the structure is kept while projecting it with $f$ but why it is not enough to check if the function is additive?

I would be glad to have some easy examples and an intuition why we would have to check both conditions.

PythonSage
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killertoge
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  • My intuition is: f(ka)=f(a+...+a)= f(a)+...+f(a)=kf(a), so if a function is additive, it MUST be homogenous. Where am I thinking wrong? – killertoge Jan 09 '21 at 15:50
  • That argument only works if $k\in \mathbb Q$. You can't get all of $\mathbb R$ that way. – lulu Jan 09 '21 at 15:54
  • For that reason: if you add continuity (or some similar property) you can get the rest of the way. You'll see from the linked duplicate that the counterexamples are wildly discontinuous. – lulu Jan 09 '21 at 15:55
  • Ok, thank you. Brings me closer to understand that duplicate thread – killertoge Jan 09 '21 at 16:01

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