Basis and Dimension — From Grassmann to van der Waerden

Abstract

Basis and dimension are two elementary notions in the theory of vector spaces. The origin of the term ‘basis’ comes from the possibility of expressing any element of a given set as a linear combination of the basis elements. Therefore, the origin lies in a question of generation; on the other hand the condition of unicity brings out the question of independence. The connection between generation and dependence is certainly one of the most interesting characteristics of the concept of basis: any maximal set of independent vectors or any minimal set of generators, is a set of independent generators and vice versa and such a set is a basis. Moreover, the dimension, beyond its “natural meaning”, is the merging point from which the question of invariance is to be drawn out. Indeed, the fact that all bases have the same number of elements entails two results: there cannot be more than a certain number of independent vectors, and fewer than the same number of generators. With a suitable starting point in the presentation of definitions and first properties on dependence and generation, these different aspects seem quite logically connected and easily explainable, but historically, the development of these two concepts was less straight-forward. For various reasons, in the approach to the concept of basis, the connections between dependence and generation were not always exhibited. Therefore the concept of dimension could only partially be drawn out, and some of its aspects were smothered, or even considered as obvious and assumed to be true without proof. On the other hand, the relation between the dimension of a subspace and the rank of any system of linear equations by which it can be represented, played a role in the history of the concepts of basis and dimension.