Basis: Difference between revisions

Line 1:

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

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/).

Line 7:

=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 {{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}}.

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 {{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.

=Mathematical details=

Line 15:

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

The vectors that appear explicitly in a basis are called the '''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.

Line 49:

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 {{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 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".

[[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.
+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/).
@@ Line 7: / Line 7: @@
 =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 {{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}}.
-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 {{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.
 =Mathematical details=
@@ Line 15: / Line 15: @@
 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 "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 explicitly listed vectors of a basis are called '''basis vectors'''.
+The vectors that appear explicitly in a basis are called the '''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.
@@ Line 49: / Line 49: @@
 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 {{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 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".
 [[Category:Regular temperament theory]]
 [[Category:Terms]]
 [[Category:Math]]