Multivariate Vector Linear Functions

Definition

A function $f: \R ^n \to \R ^m$ is linear if

$f(x + y) = f(x) + f(y)$ for all $x, y \in \R ^n$, and
$f(\alpha x) = \alpha f(x)$ for all $\alpha \in \R $ and $x \in \R ^n$.

Equivalently, $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$ for all $\alpha , \beta \in \R $ and $x, y \in \R ^n$. In this case, some authors say that superposition holds for $f$.