Basis: Difference between revisions

Line 1:

~~#redirect~~ [[~~Generator#Generators~~ in ~~math~~ and ~~JI subgroups~~]]

A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them.

The plural of "basis" is "bases" (pronounced BAY-sees, or /ˈbeɪ siz/).

Bases are mathematical structures that come from the field of [[Wikipedia:Linear algebra|linear algebra]], and are used in [[regular temperament theory]], where the most common example of a basis is a [[comma basis]]. The fact that a comma basis is a ''basis'' conveys how when a temperament [[tempers out]] the set of commas explicitly listed in a comma basis, then it also tempers out any interval that's equal to any combination of those commas. We could never possibly list the infinitude of commas tempered out, so instead we carefully choose a minimal set of commas that is capable of representing all of them.

=Examples=

For example, the comma basis {{bra|{{vector|4 -4 1}}}} does not only have a single member, {{vector|4 -4 1}}. It also contains {{vector|8 -8 2}}, {{vector|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector|-4 4 -1}}.

The comma basis {{bra|{{vector|4 -4 1}} {{vector|7 0 -3}}}} doesn't merely include {{vector|4 -4 1}} and {{vector|7 0 -3}}; it also includes {{vector|4 -4 1}} + {{vector|7 0 -3}} = {{vector|11 -4 -2}}, and 2·{{vector|4 -4 1}} + -1·{{vector|7 0 -3}} = {{vector|1 -8 5}}, and many many more.

=Mathematical details=

In mathematical language, a [[Wikipedia:Basis_(linear_algebra)|basis]] for a [[Wikipedia:Linear_subspace|subspace]] of a [[Wikipedia:Vector_space|vector space]] is a minimal set of [[Wikipedia:Vector_(mathematics_and_physics)|vectors]] that [[Wikipedia:Linear_span|span]] the subspace.

The larger set of vectors it represents is called the '''subspace'''. A mathematical word for the set of all commas tempered out by a temperament is a "nullspace", and specifically this is the nullspace of its [[mapping]] matrix; "nullspace" uses the word "space" in this same sense of a "subspace".

The explicitly listed vectors of a basis are called '''basis vectors'''.

The verb used for the process by which linear combinations of the basis vectors reach all of the subspace vectors is "spanning"; we say that the basis vectors '''span''' the subspace.

Importantly, a set of vectors that spans a subspace but is not [[full-rank|full-grade]], that is, includes [[linear dependence|linearly dependent]] vectors, or in less technical terms "redundant" vectors, is not considered a basis; in that case, it is merely a spanning set.

==Relationship to groups==

Bases are a concept in vector spaces, the subject of linear algebra. The analogous concept for [[Wikipedia:Group_(mathematics)|groups]] (and [[Wikipedia:Module_(mathematics)|modules]]), which are more general structures within the broader field of [[Wikipedia:Abstract_algebra|abstract algebra]], is a [[Wikipedia:Generating_set_of_a_module|minimal generating set]].

{| class="wikitable"

|+

!"Within a {}, ..."

!"...a {}..."

!"...consists of {}..."

!"...which {}..."

!"...a {}."

|-

|vector space

|basis

|basis vectors

|span

|subspace

|-

|group

|minimal generating set

|generators

|generate

|subgroup

|}

The sense of "subgroup" in this table is different than [[Just_intonation_subgroup|the specialized meaning it has taken on in RTT]]. Also, the sense of "generator" in this table is different than the one used for [[MOS scale]]s in the context of [[period]]s; for further disambiguating information, see [[generator]].

[[Category:Regular temperament theory]]

[[Category:Terms]]

[[Category:Math]]

@@ Line 1: / Line 1: @@
-#redirect [[Generator#Generators in math and JI subgroups]]
+A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them.
+The plural of "basis" is "bases" (pronounced BAY-sees, or /ˈbeɪ siz/).
+Bases are mathematical structures that come from the field of [[Wikipedia:Linear algebra|linear algebra]], and are used in [[regular temperament theory]], where the most common example of a basis is a [[comma basis]]. The fact that a comma basis is a ''basis'' conveys how when a temperament [[tempers out]] the set of commas explicitly listed in a comma basis, then it also tempers out any interval that's equal to any combination of those commas. We could never possibly list the infinitude of commas tempered out, so instead we carefully choose a minimal set of commas that is capable of representing all of them.
+=Examples=
+For example, the comma basis {{bra|{{vector|4 -4 1}}}} does not only have a single member, {{vector|4 -4 1}}. It also contains {{vector|8 -8 2}}, {{vector|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector|-4 4 -1}}.
+The comma basis {{bra|{{vector|4 -4 1}} {{vector|7 0 -3}}}} doesn't merely include {{vector|4 -4 1}} and {{vector|7 0 -3}}; it also includes {{vector|4 -4 1}} + {{vector|7 0 -3}} = {{vector|11 -4 -2}}, and 2·{{vector|4 -4 1}} + -1·{{vector|7 0 -3}} = {{vector|1 -8 5}}, and many many more.
+=Mathematical details=
+In mathematical language, a [[Wikipedia:Basis_(linear_algebra)|basis]] for a [[Wikipedia:Linear_subspace|subspace]] of a [[Wikipedia:Vector_space|vector space]] is a minimal set of [[Wikipedia:Vector_(mathematics_and_physics)|vectors]] that [[Wikipedia:Linear_span|span]] the subspace.
+The larger set of vectors it represents is called the '''subspace'''. A mathematical word for the set of all commas tempered out by a temperament is a "nullspace", and specifically this is the nullspace of its [[mapping]] matrix; "nullspace" uses the word "space" in this same sense of a "subspace".
+The explicitly listed vectors of a basis are called '''basis vectors'''.
+The verb used for the process by which linear combinations of the basis vectors reach all of the subspace vectors is "spanning"; we say that the basis vectors '''span''' the subspace.
+Importantly, a set of vectors that spans a subspace but is not [[full-rank|full-grade]], that is, includes [[linear dependence|linearly dependent]] vectors, or in less technical terms "redundant" vectors, is not considered a basis; in that case, it is merely a spanning set.
+==Relationship to groups==
+Bases are a concept in vector spaces, the subject of linear algebra. The analogous concept for [[Wikipedia:Group_(mathematics)|groups]] (and [[Wikipedia:Module_(mathematics)|modules]]), which are more general structures within the broader field of [[Wikipedia:Abstract_algebra|abstract algebra]], is a [[Wikipedia:Generating_set_of_a_module|minimal generating set]].
+{| class="wikitable"
+|+
+!"Within a {}, ..."
+!"...a {}..."
+!"...consists of {}..."
+!"...which {}..."
+!"...a {}."
+|-
+|vector space
+|basis
+|basis vectors
+|span
+|subspace
+|-
+|group
+|minimal generating set
+|generators
+|generate
+|subgroup
+|}
+The sense of "subgroup" in this table is different than [[Just_intonation_subgroup|the specialized meaning it has taken on in RTT]]. Also, the sense of "generator" in this table is different than the one used for [[MOS scale]]s in the context of [[period]]s; for further disambiguating information, see [[generator]].
+[[Category:Regular temperament theory]]
+[[Category:Terms]]
+[[Category:Math]]