A function $L: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is defined as linear if:
- $L(\mathbf{x+y}) = L(\mathbf{x}) + L(\mathbf{y})$
- $L(c\mathbf{x}) = cL(\mathbf{x})$, $c\in\mathbb{R}$
My question is whether property 2 is necessary or if it is derivable from 1? Or if there are functions such that property 1 is satisfied but not 2. For example, if we switch $\mathbb{R}$ to $\mathbb{C}$, it's easy to see that the conjugate function satisfies 1 but not 2.
