On the purpose of nonstandard basis vectors 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?
 A: For one example, in solid state physics, the atoms of a metal or semiconductor are rarely arranged in a nice, simple orthonormal lattice.  The lattice of atoms/molecules is almost always sheared in some respect or another.  It makes far more sense to study waves and electron distributions and what have you in such a lattice using a basis natural to the lattice, instead of trying to shoe horn the lattice arrangement into an orthonormal basis.
A: About a year ago I saw a presentation on "non-orthogonal bases" which I was told are sometimes called "frames" in that context.
The main theme was that "you don't always have to spend time paring down to an orthonormal basis" and "sometimes it's beneficial to have the redundancy in the set of basis vectors." They gave an example which unfortunately I can't remember :/
You can see, though, that skipping part of the orthonormalization process would be a definite gain in computational speed. Orthogonality is highly restrictive, and forces each basis member to be responsible for an entire direction, whereas nonorthonormal frames allow some overlap.
I think the wiki page may be a useful  place to start.
