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For those who are interested, this sort of algebra would be known as the [Grassman algebra or the exterior algebra](https://en.wikipedia.org/wiki/Exterior_algeb
by BlackFly 5y ago
For those who are interested, this sort of algebra would be known as the [Grassman algebra or the exterior algebra](https://en.wikipedia.org/wiki/Exterior_algebra https://en.wikipedia.org/wiki/Exterior_algebra). It becomes much more interesting if you use non-orthonormal bases (or non-euclidean geometry), since then you need to introduce a dual basis and distinguish between contravariant vectors and covariant vectors. When you add derivatives to the mix you end up in differential geometry.
- LotusFunctor 5y agoGrassmann algebra is a very important part of it, in fact you can reconstruct it in geometric algebra. More generally though, this algebra would be known as Clifford algebra.
- mathgenius 5y agoYes, this is exterior algebra. It's also interesting to figure out how this works in ambient dimensions other than three. The author has a table of grades: 0 for scalars, 1 for vectors, 2 for "bivectors", 3 for "trivectors", and they count the number of bases for each of these grades as 1 3 3 1. These basis counts are the dimensions of the (vector space of) scalars, vectors, "bivectors", "trivectors". If you go to two ambient dimensions you get 1 2 1, and if you go to four ambient dimensions you get 1 4 6 4 1. It's Pascal's triangle.
- LotusFunctor 5y agoPascal's triangle, and also with a transparently power-set flavor to it :)