Mike's lecture on vector spaces and dual spaces: Difference between revisions
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[0] - Technically, monzos don't form a vector space but a "'''Z'''-module", | [0] - Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. One difference is that modules are not necessarily free, unlike vector spaces, and we want temperaments to be free '''Z'''-modules. (For practical purposes, this difference just means that we have to be careful not to temper out a power of a comma, or take a wedgie out of a [[contorsion|contorted]] val like 24p in the 5-limit.) | ||
[1] - Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "'''bracket product'''", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with. | [1] - Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "'''bracket product'''", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with. | ||