I was requested to show whether the simply linear transformation $f: \mathbb{C} \rightarrow \mathbb{C}$, defined as $f(z)=iz$, is $\mathbb{R}$-linear or $\mathbb{C}$-linear. However, I was never given the definition of $\mathbb{K}$-linearity over a field $\mathbb{K}$.
We know a function $f :V \rightarrow W$, with $V, W$ vector spaces over the field $\mathbb{K}$, is linear if and only if
$$f(c\alpha+\beta)=cf(\alpha)+f(\beta)$$
for $c \in \mathbb{K}, \alpha, \beta \in V$. So I assume $\mathbb{K}$-linearity refers only to the fact that this property holds for $c \in \mathbb{K}$?
In that sense, to evaluate the $\mathbb{R}$-linearity of $f$, we should evaluate whether the following property holds:
$$f(xz_1+z_2)=xf(z_1)+f(z_2) \tag{$x \in \mathbb{R}, z_i \in \mathbb{C}$}$$
At the same time, to evaluate for $\mathbb{C}$-linearity, one should inspect whether the following holds:
$$f(wz_1+z_2)=wf(z_1)+f(z_2) \tag{$w, z_i \in \mathbb{C}$}$$
In short, is it correct to say $\mathbb{K}$-linearity refers to the fact that some transformation $f$ is linear with respect to the scalars in $\mathbb{K}$?