Mathematics > Rings and Algebras
[Submitted on 24 Apr 2024]
Title:Characterization of vector spaces by isomorphisms
View PDFAbstract: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.
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
Connected Papers (What is Connected Papers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.