Ordered vector spaces

This work was motivated by Problem 49-4 from “IMAGE” Issue Number 49, p. 52, Fall 2012 which was proposed by Rajesh Pereira. Seeking the solution to the problem led to the discovery of some interesting results on ordered vector spaces.It is shown that up to an isomorphism, there is only one order on a finite dimensional real vector space of dimension greater than or equal to 2, namely a lexicographic order on the coefficient vectors with respect to certain special bases. These special bases are called non-Archimedean bases. By examining linear preservers on an ordered vector space a solution to problem 49-4 is found as a by-product.