Part of the Poincaré Project.
To understand the metric more we need to talk about vectors but to understand vectors, at least in our context, we need to talk about linear spaces.
Remember right at the start of the Poincaré Project, we talked about adding structure to sets. Like most mathematical objects with "space" in their name, a linear space is a set of objects with a particular structure added to the set consisting of relationships between and operations on the objects.
Informally, a linear space adds just enough structure to a set that the elements can be added and scaled. And that's the principal way you should think about them here.
More formally, addition of the elements must be defined in a way that is associative and commutative and such that there exists an identity element and, for each element, an inverse. This makes linear spaces commutative groups.
Further more, there is a notion of scalar multiplication (for our purposes by a real number) such that (if a and b are real numbers and u and v elements of the linear space):
a ( u + v ) = a u + a v
(a + b) v = a v + b v
a (b v) = (ab) v
1 v = v
It is very important to note what the structure of linear spaces by itself does not provide. There is no notion of the length of an element or the angle between two elements. Neither is it possible to multiply elements by each other. They can just be added and scaled.
Often linear spaces are called "vector spaces" and their elements "vectors" but, as we will soon see, it's useful for our purposes to keep "vector" to a more narrow sense.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.