At the end of chapter 1 we showed that when F was a field, the η-fold product set Fn had an addition operation defined on it, which was induced from addition in F, so that (Fn, +) became an abelian group with zero O. Moreover we were able to define a product .:F x Fn → Fn which took (α,x) to a new element of Fn called (αx). Elements of Fn are known as vectors, and elements of F scalars. The properties that we discovered in Fn characterise a vector space.