I'm studying Projective Spaces, I've collected a few books and most of them define Projective Spaces in terms of Vector Spaces, that is, they define a 'projective space structure" in the vector space and goes on developing the theory...
But if you look on the axioms of Projective Geometry in any book there is no mention of vector spaces whatsoever, here is my
question: Are there examples of Projective Spaces other than Vector Spaces?
The way it is presented in my books it looks like the theory of Projective Spaces and Linear Algebra are "isomorphic" theories.
Maybe this answer isn't completely satisfactory, but we can define a projective space in terms of modules over a division ring.
Let $A$ be a division ring and let $M$ be an $A$-module. The projective space over $M$ is $$ \Bbb P(M)=M\setminus\{0\}/\sim $$ where $x\sim y$ if and only if there exists a $\lambda\in A^*$ such that $x=\lambda\cdot y$.