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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?

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Perhaps you will soon do things like "diagonalize a matrix" ... which means choosing a basis to make the matrix of a linear transformation as simple as possible. – GEdgar Mar 19 '13 at 18:46
Often there is a basis that is particularly suitable for an application. For example, with 3D objects there are some 'natural' bases, eg, one that fixes a point on the surface of the earth and one viewed from the orientation of the object itself. Another use is that some bases such as orthonormal bases are particularly easy to compute with. Other areas such as Fourier series can be viewed as a change of basis. – copper.hat Mar 19 '13 at 18:47
I can't tell if you are distinguishing basis vectors from their coordinates. They are often confused. If $v_1\dots v_n$ is any basis of $F^n$, then the coordinates of the basis vectors are given by "standard vectors." Among these bases is the "standard basis." Among the other suggestions already given, I think that working with these nonstandard bases would lend you more confidence with this distinction. – rschwieb Mar 19 '13 at 19:00
If you are standing somewhere on planet Earth, don't you think you might, at least occasionally, have some use for a basis consisting of vectors pointing North, East and Up. As opposed to a standard basis centered down at the center of Earth, pointing to the North pole and two fixed points on the equator? Furthermore, you may move from one point on the surface to the other, say you are piloting an airliner, and need to know how to switch from one basis to another. – Jyrki Lahtonen Mar 19 '13 at 20:00
up vote 3 down vote accepted

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.

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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 remmeber :/

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 googled around a bit, and found this presentation which I don't believe is the speaker I heard, but who appears to be talking about the same ideas.

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