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> Similarly, to rotate vectors you have to create matrices, which don't exist in ℝ2, and apply them to vectors through matrix multiplication. You don't "need"
by ajamesm 10y ago
> Similarly, to rotate vectors you have to create matrices, which don't exist in ℝ2, and apply them to vectors through matrix multiplication.
You don't "need" matrices, they're just one way to represent linear functions that map vectors, and the complex field is another (for R^2).
Coincidentally, most of the concrete implementation described for operations in G^2 here are applications of Euler's formula.
This article drops off in extending the lessons from R^2 to R^3. Comparatively, matrices excel for comprehensibility in this regard -- if you know how to apply matrices in R^N, you can apply the same knowledge to R^N+1.
- klodolph 10y agoIt's a reasonable nit, but that section is just background and rotors are introduced farther down. Rotors are unit elements in the even-numbered subspace of Cl_2(R), and that subspace can be shown to be (ring) isomorphic to the complex numbers. Personally I find the formulations for geometric algebra to be simpler than the formulations for equivalent concepts using matrices and linear algebra, in the situations where they exist.
- Koshkin 10y ago> You don't "need" matrices, they're just one way to represent linear functions Is there another way?
- ianai 10y agoAs vectors, as algebraic formulae, as a graph, etc