Mike's lecture on vector spaces and dual spaces: Difference between revisions
m →1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACE.: technically monzos form a Z-module, not a vector space |
m add note on vector spaces vs modules |
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==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''.== | |||
==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE''' | |||
For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors: | For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors: | ||
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[0] - Technically, monzos don't form a vector space but a "'''Z'''-module", becaus '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. However, the difference between modules and vector spaces don't matter to us here. (For practical purposes, this difference just means that we have to watch out for tempering out a power of a comma, or taking 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. | ||