Basis: Difference between revisions

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~~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/).

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 ~~[[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}}~~}} 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}}.

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}}~~}} 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.

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 ==

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.

~~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 "~~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 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 explicitly ~~listed vectors of~~ a basis are called '''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.

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 ===

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"

|+

|+ Comparison of terms in linear algebra and group theory

!"Within a {}, ~~...~~"

|-

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

! "Within a {}, …"

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

! "…a {}…"

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

! "…consists of {}…"

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

! "…which {}…"

! "…a {}."

|-

|vector space

| vector space

|basis

| basis

|basis vectors

| basis vectors

|span

| span

|subspace

| subspace

|-

|group

| group

|minimal generating set

| minimal generating set

|generators

| generators

|generate

| generate

|subgroup

| 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]].

== Basis vs subspace ==

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 [{{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".

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:Terms]]

[[Category:Math]]

@@ Line 1: / Line 1: @@
-A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them.
+{{Wikipedia| Basis (linear algebra) }}
-The plural of "basis" is "bases" (pronounced BAY-sees, or /ˈbeɪ siz/).
+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 [[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}}}} 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}}.
+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}}}} 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.
+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 ==
-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.
-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 "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 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 explicitly listed vectors of a basis are called '''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.
-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 ===
+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"
-|+
+|+ Comparison of terms in linear algebra and group theory
-!"Within a {}, ..."
+|-
-!"...a {}..."
+! "Within a {}, …"
-!"...consists of {}..."
+! "…a {}…"
-!"...which {}..."
+! "…consists of {}…"
-!"...a {}."
+! "…which {}…"
+! "…a {}."
 |-
-|vector space
+| vector space
-|basis
+| basis
-|basis vectors
+| basis vectors
-|span
+| span
-|subspace
+| subspace
 |-
-|group
+| group
-|minimal generating set
+| minimal generating set
-|generators
+| generators
-|generate
+| generate
-|subgroup
+| 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]].
+== Basis vs subspace ==
+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 [{{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".
+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:Terms]]
 [[Category:Math]]