Linear Algebra – Map Closed Under Addition but Not Multiplication

Question

I have been helping undergrads in an introduction to linear algebra course. When solving some exercise consisting in showing that a map is linear some get lazy after proving that it is closed under addition and do not prove the closure under scalar multiplication. I wanted to confront them with an example of a map closed under addition but not under the multiplication but could not come up with an example. Do you have any?

Answer 1

$T:\mathbb{C}\to\mathbb{C}$ defined by $T(z)=\bar{z}$ then $T(z_1+z_2)=T(z_1)+T(z_2)$ but $T(cz)\neq cT(z)$