Basis: Difference between revisions
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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. | ||
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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 | 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 === | ||
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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#The steps|Linear-dependence basis]] | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||