Say we have a linear map, $f: \mathbb{R}^{m+1} \to \mathbb{R}^{n+1}$, and we define $\mathbb{RP}^{n}$ as $(\mathbb{R}^{n+1} - \{0\})/{\sim}$ with $\sim$ define by $x \sim y$ if $y = \lambda x$ for some $\lambda \neq 0 \in \mathbb{R}$.
Then, if we define $[f]$ as $[f]:\mathbb{RP}^{n} \to \mathbb{RP}^{m}$; $[x] \mapsto [f(x)]$ what is a necessary and sufficient condition for this to define a map?
Now, after looking at this, I thought it would just be that $f$ is linear, but that can't be it, since $f$ is stated to be linear. Can anyone clarify this for me?