$\mathbb{R}$-linear and $\mathbb{C}$-linear What does it mean for a map $\phi$ to be $\mathbb R$-linear or $\mathbb{C}$-linear? I have tried to look for this definition on Google but I cannot find it.

For example, $\mathbb C$ is both a real vector space and a complex vector space.  The "complex conjugate" map $z \mapsto \overline{z}$ is real-linear but not complex-linear.  
explanation
Of course it is additive:
$$
\overline{z+w} = \overline{z}+\overline{w}
$$
and real scalars are OK
$$
\overline{rz} = r\overline{z}\qquad\text{if $r$ is real}
$$
But complex scalars are sometimes not OK, for example
$$
\overline{iz} = i\overline{z}
$$
is false when $z \ne 0$.
