I'm in a course on linear algebra right now, and I have noticed we constantly do examples and problems with nonstandard basis vectors.
I understand that we can often convert from one basis to another, and if the dimensions of the domain and co-domain are the same there is a unique linear transformation from one to the other.
My question is what is the purpose of studying these nonstandard basis vectors?
It seems that all the applications of basis vectors would use a basis which of the form $e_1 = (1,0,....)$ $e_2 = (0,1,0,..)$ etc. as this is simply the most natural way to describe coordinates in a basis.
Are there other uses for different basis' or is this just an abstract extention of the mathematics?