Characterization of vector spaces by isomorphisms

Abstract

A vector space is commonly defined as a set that satisfies several conditions related to addition and scalar multiplication. However, for beginners, it may be hard to immediately grasp the essence of these conditions. There are probably a fair number of people who have wondered if these conditions could be substituted with ones that seem more straightforward. This paper presents a simple characterization of a finite-dimensional vector space, using the concept of an isomorphism, aimed at readers with a fundamental understanding of linear algebra. An intuitive way to understand an $N$-dimensional vector space would be to perceive it as a set (equipped with addition and scalar multiplication) that is isomorphic to the set of all column vectors with $N$ components. The method proposed in this paper formalizes this intuitive understanding in a straightforward manner. This method is also readily extendable to infinite-dimensional vector spaces. While this perspective may seem trivial to those familiar with algebra, it may be useful for those who have just started learning linear algebra and are contemplating the above question. Moreover, this approach can be generalized to free semimodules over a semiring.