Topology calculation help

Let $\mathbb{RP}^n= ( \mathbb{R}^{n+1} - \{ 0 \} ) / \sim$ where $x \sim y$ if $y=\lambda x, \lambda \neq 0 \in \mathbb{R}$ adn the equivalence class of $x$ is denoted $[x]$.

what is the necessary and sufficient condition on the linear map $f : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{m+1}$ for the formula $[f][x]=[f(x)]$ to define a map

$[f] : \mathbb{RP}^n \rightarrow \mathbb{RP}^m ; [x] \mapsto [f(x)]$

not a scooby.