Why is $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto\overline{z}$ a linear map?

Question

I'm trying to understand the concept of linear maps right now. Can anyboy be so kind and perhaps very briefly illustrate to me why $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto\overline{z}$ is a linear map and where the difference is between $K=\mathbb{R}$ and $K=\mathbb{C}$?

Answer 1

Clearly, $f(z_1+z_2)=\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}=f(z_1)+f(z_2)$. And as long as $\lambda\in \Bbb R$, i.e., $\overline\lambda=\lambda$, we also have $f(\lambda z)=\overline{\lambda z}=\overline\lambda\overline z=\lambda\overline z=\lambda f(z)$. Hence $f$ is $\Bbb R$-linear. But it is not $\Bbb C$-\linear since we see for example with $\lambda=i$ and $z=1$ that $f(\lambda z)\ne\lambda f(z)$.