Is there an element $f$ of $\operatorname{End}(V)$ which is not an element of $\operatorname{End}_K(V)$?

Question

I am reading "An Introduction to Algebraic Systems (in Japanese) by Kazuo Matsuzaka.

Let $V$ be a vector space over a field $K$.
Let $\operatorname{End}_K(V)$ be the set of all linear mappings from $V$ to $V$.
Let $\operatorname{End}(V)$ be the set of all homomorphisms from the additive group $V$ to $V$.

The author wrote $\operatorname{End}_K(V)$ is a subring of $\operatorname{End}(V)$.

Is there an element $f$ of $\operatorname{End}(V)$ which is not an element of $\operatorname{End}_K(V)$?

Answer 1

$\newcommand{\C}{\Bbb C} \newcommand{\End}{\operatorname{End}}$A classic example is the following:

The function $\C \to \C$ that sends $z \in \C$ to $\bar z$ (its complex conjugate) is an element of $\End(\C)$ (because $$\overline{z_1+z_2} = \overline{z_1}+\overline{z_2}$$ for any two $z_1,z_2 \in \C$) but not an element of $\End_\C(\C)$ (for example, it is not true that $$ \overline{iz} = i \bar{z}$$ for $z=i$).