Definition of a Linear Mapping

Let $V$ and $W$ be $K$-vector spaces. A mapping

\[

f : V \to W

\]

is called \textit{linear} if

i)	$f(x + y) = f(x) + f(y)$ for all $x, y \in V$,
ii)	$f(\lambda x) = \lambda f(x)$ for all $\lambda \in K$ and $x \in V$.