What's the meaning of that $f: U\to \Bbb R^n$ is linear?

Question

Let $U$ be an open set of $\Bbb R^m$. What's the meaning of that a map $f: U\to \Bbb R^n$ is linear?

Answer 1

$$f: \Bbb R^m\to \Bbb R^n$$ is linear if and only if for all $\alpha, \beta, V_1, V_2$ we have $$ f(\alpha V_1 + \beta V_2) = \alpha f(V_1) + \beta f(V_2)$$

Such functions are also called linear transformations between two vector spaces.