Basis: Difference between revisions
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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 | 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 | 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 | 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 | 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 | 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". | ||