Coordinate Systems
Points are positions in space. When first introduced in Greek geometry, there was no formal method for measuring where a point was located. Points were primitive entities. Much later in history, in the early seventeenth century, Fermat and Descartes introduced the idea of using a linear coordinate system to specify point locations, using algebraic equations to describe geometric objects (such as lines), and using algebra to solve geometric problems (such as computing the intersection point of two lines).
A coordinate system has an origin as an absolute reference point whose location is fixed, and nonzero axes (a set of direction vectors) that determine the directions in which to make measurements. A coordinate system can specify points in ndimensional space as a ordered ntuple of numbers and the numeric rule: start at the origin, go distance x_{1} in the direction of the first axis, stop, now go distance x_{2} in the direction of the second axis, stop, and so on until done with x_{n}. The final location that one reaches is the point specified. This is much like finding the treasure on a pirate's map; for example, start at the old oak tree as the origin, first go east 10 paces, second go north 20 paces, and third dig down 3 feet to locate the treasure. The ordered set of numbers is called the coordinate of the point it specifies. And a coordinate system is said to span the set of points that it can specify. The set of all these points is called the coordinate space. A specific set of axes spanning a coordinate space is called a coordinate frame of reference or a basis for the space. Clearly, different coordinate systems can span the same coordinate space, just as the pirate may have used any of several origin trees or direction axes for his treasure map.
The axes are said to be independent if they do not depend on each other, meaning that none of them can written as a combination of the others. When independent, they are a minimal set of axes for the coordinate space, and the number of axes in the set is the dimension of the space.
