Real Linear Transformations

Definition

A real linear transformation is a function $f: \R ^n \to \R ^m$ satisfying

\[ f(\alpha x + \beta y) = \alpha f(x) + \beta f(y) \quad \text{for all } x, y \in \R ^n \text{ and } \alpha , \beta \in \R \]

Equivalently, $f$ is (a) homogenous $f(\alpha x) = \alpha f(x)$ and (b) additive $f(x + y)= f(x) + f(y)$.