Rank and codimension: Difference between revisions

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The thing called "codimension" above can be interpreted in linear algebra terms as the codimension of the subspace of supporting vals, relative to the ambient space of all vals. For any mapping matrix mapping from monzos to tmonzos, it's also the co-rank of the dual transformation from tvals back to vals.

==Rank-nullity theorem==

The [https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem the rank-nullity theorem] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.

All three of rank, nullity, and dimensionality are types of dimension:

* Rank is the dimension of the image (or range).

* Nullity is the dimension of the null-space (or kernel).

* Dimensionality is the dimension of the domain. It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".

Collectively we could refer to these as a temperament's '''dimensions'''.

The rank-nullity theorem is also discussed in [[Mike's Lecture on the First Fundamental Law of Tempering]].

[[Category:Regular temperament theory]]

[[Category:Temperament]]

[[Category:Theory]]

[[Category:Terms]]

[[Category:Math]]

@@ Line 19: / Line 19: @@
 The thing called "codimension" above can be interpreted in linear algebra terms as the codimension of the subspace of supporting vals, relative to the ambient space of all vals. For any mapping matrix mapping from monzos to tmonzos, it's also the co-rank of the dual transformation from tvals back to vals.
+==Rank-nullity theorem==
+The [https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem the rank-nullity theorem] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.
+All three of rank, nullity, and dimensionality are types of dimension:
+* Rank is the dimension of the image (or range).
+* Nullity is the dimension of the null-space (or kernel).
+* Dimensionality is the dimension of the domain. It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".
+Collectively we could refer to these as a temperament's '''dimensions'''.
+The rank-nullity theorem is also discussed in [[Mike's Lecture on the First Fundamental Law of Tempering]].
 [[Category:Regular temperament theory]]
+[[Category:Temperament]]
+[[Category:Theory]]
+[[Category:Terms]]
 [[Category:Math]]