Let $k$ be a field and let $V$ and $W$ be finite-dimensional $k$-vector spaces, where $\dim(V)\ge1$ and $\dim(W)\ge1$. Let $q:V\to\mathbb{P}(V)$, $u\to[u]$ be the quotient map. By my teacher, a map $f:\mathbb{P}(V)\to\mathbb{P}(W)$ is defined to be a projective map if there exists an injective linear map $L:V\to W$ such that for all $[p]\in\mathbb{P}(V)$ we have that $f([p])=A(P)$, where $P$ is the subspace spanned by any $p\in V$ for which $q(p)=[p]$ holds. Consequently, he defines a projective transformation to be a projective map $f:\mathbb{P}(V)\to\mathbb{P}(V)$. These projective transformations, as I understand it, form Aut$(\mathbb{P}(V))$.
That's all nice and well, but I was wondering if this really is the most "elementary" way of looking at Aut$(\mathbb{P}(V))$, and if it really is necessary to define what an element of Aut$(\mathbb{P}(V))$ looks like. It now seems so "derived" from Aut$(V)$. I hope that my point is coming across.
For example, when we define a group, ring, $R$-module, topological space homomorphism it is just a map (of sets) which has the structure-preserving properties you want, but these are not "derived" from maps on some other category, such as it seems to be here, as this definition implies the surjection Aut$(V)\to$Aut$(\mathbb{P}(V))$. We don't say (or at least I have not seen it) that any group homomorphism for example has to come from I don't know, a map of sets.
If someone can clears this up even a little bit that would be great, and I apologise for any basic errors which someone like me who hasn't formally and rigorously done category theory (but is fascinated by the concepts introduced by it) yet.
You're right to worry about this; in more general settings ($k$ not a field but a more general commutative ring) the natural map $\text{Aut}(V) \to \text{Aut}(\mathbb{P}(V))$ might fail to be surjective, so it won't be possible to use this definition. Unfortunately, the correct definition at this level of generality is relatively sophisticated (even correctly defining $\mathbb{P}(V)$ itself is relatively sophisticated); your professor's definition is certainly the most elementary in that it's the one that requires developing the least additional material to use.