Norms, Isometries, and Isometry Groups

The study of linear algebra has become more and more popular in the last few decades. People are attracted to this subject because of its beauty and its connections with many other pure and applied areas. In theoretical development of the subject as well as in many applications, one often needs to measure the “length” of vectors. For this purpose, norm functions are considered on a vector space. In this expository article, we explain why one would want to study different kinds of norms on a real vector space. We then focus on the problem of how to identify different norms using linear isomorphisms with particular attention to a group theory method used by several authors recently. One may view the first half of this article as a gentle introduction to the theory of norms and the second half as an illustration of how group theory can be applied to questions in linear algebra.