quality:correction, structural:expansion, linguistic:paraphrasing
Euclidean vectors

Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R.Consider the vectors e = (1,0,0), e = (0,1,0) and e = (0,0,1).Then any vector in R is a linear combination of e, e, and e.

To see that this is so, take an arbitrary vector (a, a, a) in R, and write: