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 SPACETechnically, 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. For practical purposes, this technicality is completely accounted for by tools like the x31eq temperament finder --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperamen... |
m →1.1: A monzo can be viewed as a VECTOR in a VECTOR SPACE[1].: idk if slight is the right word, since Z modules not necessarily being free is a big difference! Tags: Mobile edit Mobile web edit |
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==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>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. | ==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>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. For practical purposes, this technicality is completely accounted for by tools like the [http://x31eq.com/temper/ x31eq temperament finder] --- it might tell you that your temperament is a "contorted" version of another temperament, and in temperament theory we usually want non-contorted temperaments.</ref>.== | ||
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: | ||