Abstract definition of Projection I'm coming from the programming domain and is searching the mathematical domain for a better understanding of the term Projection sometimes used in programming as well.
After reading a lot of Wikipedia sources I think I have finally gotten a good picture of how the domain looks here even if the details are way over my head.
The Wikipedia article about Projection (mathematics) gives a good example with the dot and the shadow, but I'm having problem with this statement: 

The restriction to a subspace of a projection is also called a
  projection, even if the idempotence property is lost.

What does "The restriction" refer to? Any restriction? Any subset of a projection is an projection in itself? The function that limits the projection set is itself a projection of the projection?
In the example with the dot and the shadow, how would this subset of the shadow that implies non-idempotency be achieved?
 A: There is not much to be said here. Let $P:V \to V$ be a projection; this means that $P^2 = P \circ P = P.$ If we consider the restriction of $P$ to a subset $A \subset V,$ $P|_A:A \to V,$ it does not really make sense to compose $P|_A$ with itself, since the domain and codomain are different. Hence, $P|_A$ technically isn't idempotent. Still, we say it is a projection.
A: You correctly start with the "graphical" concept of Projection.
When you project a 3D shape $S$ on the $x-y$ plane (think in term of 3D computer graphics) you are performing an operation that in terms of analytical geometry take each point $P(x,y,z) \in S$ and "transform" it into a point $P'(x,y,0)$, that obviously belongs to the $x-y$ plane.
In this way, all points of $S$ with the same couple of coordinates $(x,y)$ and different values of $z$ will "collapse" into the same point on the $x-y$ plane.
From Wiki : in the same way

in set theory the operation called $j$-th projection map, written $proj_j$ , takes an element $x = (x_1, ..., x_j , ..., x_k)$ of the cartesian product $X_1 \times … \times X_j \times … \times X_k$ to the value $proj_j (x) = x_j$ . 

The idempotency property means that when you have projected $S$ on the plane, a successive projection on the same plane will not "restrict" your "image" any more.
