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What are some examples of interesting interactions (that you'd like to see, not necessarily already exist) that illustrate linear algebra with non-R^n vector sp

by gowld 7y ago

What are some examples of interesting interactions (that you'd like to see, not necessarily already exist) that illustrate linear algebra with non-R^n vector spaces? Thanks to representation theory, we can model vector spaces as R^n https://en.wikipedia.org/wiki/Representation_theory https://en.wikipedia.org/wiki/Representation_theory

impendia 7y ago
I don't really think that representation theory is required here, it's easier than that. (Although representation theory is really cool!!) Every finite dimensional vector space over the real numbers is isomorphic to R^n, for some n. But there is not always a canonical or "unique" isomorphism. I think the real difference -- and it is a subtle one -- is that R^n always comes with an "obvious" choice of basis, and many vector spaces don't. I think the following may be the easiest interesting example. Consider the subspace of R^3, consisting of all (x, y, z) for which x + y + z = 0. This is a 2-dimensional real vector space. It is isomorphic to R^2, although in some sense it does not "present as such": you have to choose an isomorphism to R^2 if you want to "treat it as R^2", and there is no single choice that stands out. In practice, one would not necessarily construct an isomorphism to R^2, or (more or less equivalently) exhibit a basis, before working with this vector space directly.
kd0amg 7y ago
I think you could do some interesting stuff with error-correcting codes or polynomial spaces (or more generally function spaces). I'm not too familiar with representation theory. Do you know where I could find a worked example of constructing, say, a representation of (F_p)^m on R^n? (for prime p and m>1)
- gowld 7y ago
  Visually, (F_p)^m can be embedded in R^m trivially, but I suspect you are interested in animations of operations on F^p, which indeed would look different from R^m due to modular arithmetic, and so would be beyond the scope of the OP.

by gowld 7y ago

impendia 7y ago

I don't really think that representation theory is required here, it's easier than that. (Although representation theory is really cool!!) Every finite dimensional vector space over the real numbers is isomorphic to R^n, for some n. But there is not always a canonical or "unique" isomorphism. I think the real difference -- and it is a subtle one -- is that R^n always comes with an "obvious" choice of basis, and many vector spaces don't. I think the following may be the easiest interesting example. Consider the subspace of R^3, consisting of all (x, y, z) for which x + y + z = 0. This is a 2-dimensional real vector space. It is isomorphic to R^2, although in some sense it does not "present as such": you have to choose an isomorphism to R^2 if you want to "treat it as R^2", and there is no single choice that stands out. In practice, one would not necessarily construct an isomorphism to R^2, or (more or less equivalently) exhibit a basis, before working with this vector space directly.

kd0amg 7y ago

I think you could do some interesting stuff with error-correcting codes or polynomial spaces (or more generally function spaces). I'm not too familiar with representation theory. Do you know where I could find a worked example of constructing, say, a representation of (F_p)^m on R^n? (for prime p and m>1)

gowld 7y ago
Visually, (F_p)^m can be embedded in R^m trivially, but I suspect you are interested in animations of operations on F^p, which indeed would look different from R^m due to modular arithmetic, and so would be beyond the scope of the OP.