Linear dependence: Difference between revisions
Cmloegcmluin (talk | contribs) footnotes |
Cmloegcmluin (talk | contribs) temperament arithmetic → temperament addition |
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When basis vector sets do not share a common basis vector like this, they are '''linearly ''in''dependent'''. Linearly dependent basis vector sets are in a sense more closely related to each other than linearly independent basis vector sets. | When basis vector sets do not share a common basis vector like this, they are '''linearly ''in''dependent'''. Linearly dependent basis vector sets are in a sense more closely related to each other than linearly independent basis vector sets. | ||
Linear dependence is involved in certain operations used in regular temperament theory, such as the [[Wikipedia:Wedge_product|wedge product]] or [[temperament | Linear dependence is involved in certain operations used in regular temperament theory, such as the [[Wikipedia:Wedge_product|wedge product]] or [[temperament addition]], which are defined for objects that can be interpreted as basis vector sets, such as matrices or multivectors, and that also represent regular temperaments. | ||
== Linear dependence as defined for various types of vector sets == | == Linear dependence as defined for various types of vector sets == | ||
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=== Linear dependence between temperaments === | === Linear dependence between temperaments === | ||
The conditions of temperament | The conditions of temperament addition motivate a special definition of linear dependence for temperaments. For more information, see: [[Temperament addition#2. Linear dependence between temperaments]]. | ||
== RTT applications involving linear dependence == | == RTT applications involving linear dependence == | ||
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Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a temperament merge. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent temperament merge operation in linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]] | Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a temperament merge. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent temperament merge operation in linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]] | ||
=== Temperament | === Temperament addition === | ||
Temperament | Temperament addition only results in a usable temperament when the input temperaments are addable, an advanced property that builds upon linear dependence. For more information, see [[Temperament addition#Addability]]. | ||
== Variance == | == Variance == | ||
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Linear dependence is defined both for basis vector sets whether they are covariant ("''co''vectors", such as [[map]]s) or contravariant (plain "vectors", such as [[prime count vector]]s). For simplicity, this article will use the word "vector" in its general sense, which includes either plain/contravariant vectors or (covariant) covectors.<ref>This article will also use "multivector" to refer to either plain/contravariant multivectors or (covariant) multicovectors (elsewhere on the wiki you will find "varianced multivector" to refer unambiguously to either type in the general sense).</ref> | Linear dependence is defined both for basis vector sets whether they are covariant ("''co''vectors", such as [[map]]s) or contravariant (plain "vectors", such as [[prime count vector]]s). For simplicity, this article will use the word "vector" in its general sense, which includes either plain/contravariant vectors or (covariant) covectors.<ref>This article will also use "multivector" to refer to either plain/contravariant multivectors or (covariant) multicovectors (elsewhere on the wiki you will find "varianced multivector" to refer unambiguously to either type in the general sense).</ref> | ||
Plain vectors and covectors cannot be compared with each other, however. Linear dependence is only defined for a set of basis vector sets, or a set of basis covector sets. Linear dependence is not defined for a set including both basis vector sets and basis covector sets. For example, a set including one [[mapping]] (a basis covector set) and one [[comma basis]] (a basis "plain-vector" set) has no directly meaningful notion of linear dependence<ref>though the two temperaments here — the one defined by this mapping, and the other defined by this comma basis — can have a notion of linear dependence, as can be understood by finding the comma basis that is the dual of the mapping and checking the two comma bases for linear dependence, or vice versa, finding the mapping that is the dual of the comma basis and checking the two mappings for linear dependence. This notion of linear dependence is discussed in more detail [[Temperament | Plain vectors and covectors cannot be compared with each other, however. Linear dependence is only defined for a set of basis vector sets, or a set of basis covector sets. Linear dependence is not defined for a set including both basis vector sets and basis covector sets. For example, a set including one [[mapping]] (a basis covector set) and one [[comma basis]] (a basis "plain-vector" set) has no directly meaningful notion of linear dependence<ref>though the two temperaments here — the one defined by this mapping, and the other defined by this comma basis — can have a notion of linear dependence, as can be understood by finding the comma basis that is the dual of the mapping and checking the two comma bases for linear dependence, or vice versa, finding the mapping that is the dual of the comma basis and checking the two mappings for linear dependence. This notion of linear dependence is discussed in more detail [[Temperament addition#2. Linear dependence between temperaments|here]].</ref>. So, while it is convenient to use "vector" for either type, it is important to be careful to use only on type at a time, never mixing the two types. | ||
== Versus collinearity == | == Versus collinearity == | ||