Motivation behind definition of Projective space. Some definitions of a projective space are as follows:
First: A projective space of dimension $n$ can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension $n + 1.$.
Second: A projective space over the field $F$ is a triple $(P; E; \pi)$ where $P$ is a set, $E$ is a finite-dimensional vector space over $F$ , dim $E\geq 2$, and $\pi$ is a map, $\pi:E\setminus\{0\}\longrightarrow P$,
satisfying:
a) $\pi$ is onto, and
b) for any $v, w \in E$, $\pi(v)=\pi(w)$ if and only if there is a (necessarily non-zero)  $\lambda\in F$ such that $v=\lambda w$.
Can any one tell how these two definitions are equivalent? I know that the motivation for projective geometry comes from the notion of perspective in drawing. But how does that give rise to this definition? (A diagramatic explanation would be helpful
I have understood that the real projective plane is basically a way to give formal coordinates to the extended euclidean plane (from the book Perspectives on Projective Geometry, Richter-Gebert, Jürgen).
But I don’t understand the definition of a general $n$ dimensional projective space over any field $F$.
 A: How the two are equivalent:
In the second definition, the vector space $E$ is there to give us the vector lines from the first definition. The map $\pi$ then tells us which elements of $P$ correspond to which vector line in $E$. Specifically, if $p\in P$ is a point in the projective space, then $\pi^{-1}(p)$ is a vector line, and it is this vector line which corresponds to the point $p$. In technical terms, the second definition says that a projective space $P$ is the quotient space $(E\backslash\{0\})/\pi$ of $E\backslash\{0\}$ under the projection $\pi$ which projects all points on a vector line onto a single point. That is, every point of $P$ can be thought of as an equivalence class of vectors in $E\backslash\{0\}$, where vectors are equivalent if they belong to the same vector line. And that's just the first definition.
What this has to do with the geometric intuition:
Geometrically, the projective closure of an affine space is obtained by adding a point at infinity for every bundle of parallel lines such that every line in the bundle intersects said point. This way, parallel lines intersect "at infinity", like they do when drawing in perspective. So that's what we want to obtain in the end. The way this is usually done is by adding another dimension to the vector space whose closure we want to construct. Essentially, we embed the vector space $F^n$ in the higher dimensional space $F^{n+1}$ as an affine subspace via $(x_1,\dots,x_n)\mapsto(x_1,\dots,x_n,1)$. Let's call this subspace $A^n$. Then we identify each vector line in $F^{n+1}$ which intersects $A^n$ with the point where it intersects. So the line going through the origin and $(x_1,\dots,x_n,1)$ we identify as the point $(x_1,\dots,x_n)$ in our original vector space. But in addition to the vector lines intersecting $A^n$, we also have vector lines which lie parallel to it. These we identify as the additional points at infinity. And they do what they're supposed to: Take two parallel lines in $A^n$. The points of which they consist can be identified as vector lines. All the vector lines which make up one of the two parallel lines form a vector plane (a 2d vector subspace). Well, almost. One line is missing, and it's one of the lines parallel to $A^n$. But we can add that line to complete the vector plane, and define this vector plane as a line in the projective space. And it turns out that parallel lines in $A^n$ result in vector planes in $F^{n+1}$ which intersect at the same vector line parallel to $A^n$. That line is the point at infinity at which the original two lines intersect. Here is a diagram, the original of which can be found on Wikipedia:

The blue plane is a 2d vector space embedded into a 3d vector space as an affine subspace, and the red plane in 3d space corresponds to the red line (in the blue plane) in the affine subspace. The red plane contains the red line (now in the green plane) which is the newly added point at infinity.
