Basis: Difference between revisions

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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.

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 ===

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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.

== 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 21: / Line 21: @@
 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 ===
@@ Line 58: / Line 58: @@
 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]]