Talk:Mike's lecture on vector spaces and dual spaces: Difference between revisions
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== No vector spaces == | == No vector spaces == | ||
Also, it is not clear where a vector space comes from. A vector space is something defined over a field. But any set of intervals is not a field, it only forms an Abelian group, with * operation being the arithmetic multiplication of rational number. This is the only significant operation, as additive operation between those numbers has no meaning for music; intervals is added when a frequency is multiplied by it, and two intervals can be added or subtracted. The additive operation for frequencies is the * operation for this group. With all the similarity with vector spaces, the lack of essential vector space properties is dramatic. You can scale an interval (by an integer number), but you cannot define a | Also, it is not clear where a vector space comes from. A vector space is something defined over a field. But any set of intervals is not a field, it only forms an Abelian group, with * operation being the arithmetic multiplication of rational number. This is the only significant operation, as additive operation between those numbers has no meaning for music; intervals is added when a frequency is multiplied by it, and two intervals can be added or subtracted. The additive operation for frequencies is the * operation for this group. With all the similarity with vector spaces, the lack of essential vector space properties is dramatic. You can scale an interval (by an integer number), but you cannot define a basis, because there is no a way to decompose a vector against the basis. No, this is not a vector space at all. — [[User:SAKryukov|SA]], ''Wednesday 2020 December 2, 17:13 UTC'' | ||