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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.