Is there Any Homomorphism Between Vector Spaces that is not Linear?

Question

I am learning Abstract Algebra and professor asked the existence of group homomorphism between vector spaces that is not linear.

I think there would be one which artificially constructed since the linearity given in vector space seems independent from the property of group homomorphism.

Any example of those non-linear group homomorphism between vector spaces?

Answer 1

Consider $\mathbb C$ as a complex vector space in the usual sense. Then the conjugation is a group homomorphism from $(\mathbb{C},+)$ into itself which is not linear: $\overline{i.1}\neq i.\overline1$.

Answer 2

Perhaps your professor is referring to a morphism between two vector spaces such that:

$$f( a + b) = f(a) \oplus f(b)$$ But it happens that: $$ \lambda f(a) \neq f(\lambda a) $$

It turns out that such functions do exist. The best example of whose existence can be proved (but cannot be constructed) here.