Basis - Revision history

Cmloegcmluin: centralize set of links to RTT bases

2023-01-24T15:39:32Z

centralize set of links to RTT bases

← Older revision		Revision as of 15:39, 24 January 2023
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	We do have to be careful, though, to remember that a subspace has infinitely many possible basis representations. This is why [[canonical form]]s are typically developed for them, as they have been for mappings and comma bases, so that each subspace ''does'' have a uniquely identifying basis.		We do have to be careful, though, to remember that a subspace has infinitely many possible basis representations. This is why [[canonical form]]s are typically developed for them, as they have been for mappings and comma bases, so that each subspace ''does'' have a uniquely identifying basis.

			== See also ==

			* [[Comma basis]]
			* [[Held-interval basis]]
			* [[Domain basis]]
			* [[Unchanged-interval basis]]
			* [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D\|Linear-dependence basis]]

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]

Cmloegcmluin: full-grade doesn't make sense; should have been full-rank

2022-12-28T23:47:45Z

full-grade doesn't make sense; should have been full-rank

← Older revision		Revision as of 23:47, 28 December 2022
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	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.		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.		Importantly, a set of vectors that spans a subspace but is not [[full-rank]], 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 ===		=== Relationship to groups ===

Cmloegcmluin: add pronunciation information for plural back

2022-12-28T20:04:46Z

add pronunciation information for plural back

← Older revision		Revision as of 20:04, 28 December 2022
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	{{Wikipedia\| Basis (linear algebra) }}		{{Wikipedia\| Basis (linear algebra) }}

	A '''basis''' (plural ''bases'') is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.		A '''basis''' (plural ''bases'', pronounced BAY-sees, or /ˈbeɪ siz/) is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.

	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 makes the set of commas explicitly listed in a comma basis [[vanish]], then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.		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 makes the set of commas explicitly listed in a comma basis [[vanish]], then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.

FloraC: /* Mathematical details */ this information wasn't accurate -- basis is used in vector spaces as well as free abelian groups, which models ji

2022-12-28T13:24:49Z

Mathematical details: this information wasn't accurate -- basis is used in vector spaces as well as free abelian groups, which models ji

← Older revision		Revision as of 13:24, 28 December 2022
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	=== Relationship to groups ===		=== Relationship to groups ===
			The basis is a concept used across vector spaces and [[Wikipedia: Free abelian group\|free abelian groups]]. The analogous concept for [[Wikipedia: Group (mathematics)\|groups]] and [[Wikipedia: Module (mathematics)\|modules]] in general, which are structures within the broader field of [[Wikipedia: Abstract algebra\|abstract algebra]], is known as the [[Wikipedia: Generating set of a module\|minimal generating set]].
	~~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"		{\| class="wikitable"
	\|+		\|+ Comparison of terms in linear algebra and group theory
			\|-
	! "Within a {}, …"		! "Within a {}, …"
	! "…a {}…"		! "…a {}…"
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	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]].		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]].

	~~Unfortunately, terminology on this wiki does not consistently differentiate between these categories, and so there are occurrences of subgroups with bases, or generators for a basis, etc.~~

	== Basis vs subspace ==		== Basis vs subspace ==

FloraC: Plural needn't a separate paragraph

2022-12-28T10:07:36Z

Plural needn't a separate paragraph

← Older revision		Revision as of 10:07, 28 December 2022
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	{{Wikipedia\| Basis (linear algebra) }}		{{Wikipedia\| Basis (linear algebra) }}

	A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.		A '''basis''' (plural ''bases'') is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.

	~~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 makes the set of commas explicitly listed in a comma basis [[vanish]], then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.		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 makes the set of commas explicitly listed in a comma basis [[vanish]], then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.

Cmloegcmluin: update EBK style and terminology

2022-12-27T17:21:53Z

update EBK style and terminology

Cmloegcmluin: prefer nullspace to null-space

2022-12-15T00:16:47Z

prefer nullspace to null-space

← Older revision		Revision as of 00:16, 15 December 2022
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	In mathematical language, a 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.		In mathematical language, a 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.

	For example, a mathematical word for the set of all commas tempered out by a temperament is a "~~null-space~~", and specifically this is the ~~null-space~~ of its [[mapping]] matrix; "~~null-space~~" uses the word "space" in this same sense of a "subspace".		For example, 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 vectors that appear explicitly in a basis are called the '''basis vectors'''. More generally we can refer to these as '''basis elements'''; for example, if {{vector\|-1 1}} was a basis ''vector'', we could say <math>\frac32</math> was a basis ''element''.		The vectors that appear explicitly in a basis are called the '''basis vectors'''. More generally we can refer to these as '''basis elements'''; for example, if {{vector\|-1 1}} was a basis ''vector'', we could say <math>\frac32</math> was a basis ''element''.

Cmloegcmluin: /* Mathematical details */ basis element definition explicitly

2022-12-14T23:51:03Z

Mathematical details: basis element definition explicitly

← Older revision		Revision as of 23:51, 14 December 2022
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	For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace".		For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace".

	The vectors that appear explicitly in a basis are called the '''basis vectors'''.		The vectors that appear explicitly in a basis are called the '''basis vectors'''. More generally we can refer to these as '''basis elements'''; for example, if {{vector\|-1 1}} was a basis ''vector'', we could say <math>\frac32</math> was a basis ''element''.

	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.		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.

FloraC: +wikipedia template

2022-03-10T14:16:00Z

+wikipedia template

← Older revision		Revision as of 14:16, 10 March 2022
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			{{Wikipedia\| Basis (linear algebra) }}

	A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.		A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''.

	The plural of "basis" is "bases" (pronounced BAY-sees, or /ˈbeɪ siz/).		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.		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.
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	== Mathematical details ==		== 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.		In mathematical language, a 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.

	For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace".		For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace".

Cmloegcmluin: /* Relationship to groups */ wiki disclaimer

2022-01-09T00:02:59Z

Relationship to groups: wiki disclaimer

← Older revision		Revision as of 00:02, 9 January 2022
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	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]].		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]].

			Unfortunately, terminology on this wiki does not consistently differentiate between these categories, and so there are occurrences of subgroups with bases, or generators for a basis, etc.

	== Basis vs subspace ==		== Basis vs subspace ==

← Older revision		Revision as of 17:21, 27 December 2022
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	The plural of ''basis'' is ''bases'' (pronounced BAY-sees, or /ˈbeɪ siz/).		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.		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 makes the set of commas explicitly listed in a comma basis [[vanish]], then it also makes any interval that's equal to any combination of those commas vanish. We could never possibly list the infinitude of commas made to vanish in this way, so instead we carefully choose a minimal set of commas that is capable of representing all of them.

	== Examples ==		== Examples ==

	For example, the comma basis ~~{{bra\|~~{{vector\|4 -4 1}}}} only includes {{vector\|4 -4 1}}, but it represents the subspace that also includes {{vector\|8 -8 2}}, {{vector\|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector\|-4 4 -1}}.		For example, the comma basis [{{vector\|4 -4 1}}] only includes {{vector\|4 -4 1}}, but it represents the subspace that also includes {{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}}}} only includes {{vector\|4 -4 1}} and {{vector\|7 0 -3}}, but it represents the subspace that 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.		The comma basis [{{vector\|4 -4 1}} {{vector\|7 0 -3}}] only includes {{vector\|4 -4 1}} and {{vector\|7 0 -3}}, but it represents the subspace that 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 ==		== Mathematical details ==
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	In mathematical language, a 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.		In mathematical language, a 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.

	For example, 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".		For example, a mathematical word for the set of all commas made to vanish 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 vectors that appear explicitly in a basis are called the '''basis vectors'''. More generally we can refer to these as '''basis elements'''; for example, if {{vector\|-1 1}} was a basis ''vector'', we could say <math>\frac32</math> was a basis ''element''.		The vectors that appear explicitly in a basis are called the '''basis vectors'''. More generally we can refer to these as '''basis elements'''; for example, if {{vector\|-1 1}} was a basis ''vector'', we could say <math>\frac32</math> was a basis ''element''.
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	Subspaces and bases have a close relationship. A basis, even in its everyday dictionary definition, is an underlying support or foundation ''for something'', and in this mathematical case, that something is a subspace. Without bases, it would be much more challenging to communicate about subspaces; they're quite specific objects, but they happen to be infinitely large, and so bases were developed to be finite representations of them, for convenience.		Subspaces and bases have a close relationship. A basis, even in its everyday dictionary definition, is an underlying support or foundation ''for something'', and in this mathematical case, that something is a subspace. Without bases, it would be much more challenging to communicate about subspaces; they're quite specific objects, but they happen to be infinitely large, and so bases were developed to be finite representations of them, for convenience.

	And so it is not disingenuous to call something like 2.3.7 or ~~{{bra\|~~{{vector\|4 -4 1}} {{vector\|7 0 -3}}}} a "subspace" — if we are indeed referring to the infinitely large thing spanned by the this basis, and not the basis itself — because the entire point of bases are to enable representation of these such subspaces.		And so it is not disingenuous to call something like 2.3.7 or [{{vector\|4 -4 1}} {{vector\|7 0 -3}}] a "subspace" — if we are indeed referring to the infinitely large thing spanned by the this basis, and not the basis itself — because the entire point of bases are to enable representation of these such subspaces.

	And when we ''are'' referring to the basis itself, it's perfectly fine to refer to a "subspace basis" as a "basis" for short, as we have been doing throughout this article, because there's no other type of basis in this context; something being a "basis" here implies that it is a "subspace basis".		And when we ''are'' referring to the basis itself, it's perfectly fine to refer to a "subspace basis" as a "basis" for short, as we have been doing throughout this article, because there's no other type of basis in this context; something being a "basis" here implies that it is a "subspace basis".