Definition. Let P(V) be a projective space of dimension n. Then n+2 points in P(V) are said to be in general position if no n+1 of them are contained in a (n−1)-dimensional projective subspace. In terms of linear algebra this implies that no n+1 representative vectors are linearly dependend, i.e. every n+1 are linearly independent.
Examples. If n=1 we have to consider n+2=3 points on a projective line. These three points are in general position as long as they are disjoint.
Let us start with a general definition for an arbitrary projective space. In this lecture we will almost entirely deal with real projective spaces.
Definition. Let V be a vector space over an arbitrary field F. The projective space P(V) is the set of 1-dimensional vector subspaces of V. If dim(V)=n+1, then the dimension of the projective space is n.
The word geometry is of Greek origin (γεωμετρια) and means “measurement of the earth”. But the first accounts were found in Egypt and Mesopotamia (today Iraq) around 2000 BC. It was known how to calculate simple areas and volumes and people of that time already had an approximation of π. But no general statements/theorems and proofs were found. Around 600 BC in Greece, Thales was the first to deduce theorems from a set of axioms. At about 300 BC, Euclid wrote the book Elements in which he states the five axioms of what is today called Euclidean geometry. His fifth axioms is known as the parallel postulate and can be phrased as follows:
Given a line and a point not on this line, there is a unique line through the point parallel to the line.