Linear dependence: Difference between revisions

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[[Basis|Bases]], such as [[Comma basis|comma bases]], are considered '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]]. In other words, that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.

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.

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=== Wedge product ===

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 merging|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. Ordinarily, the wedge product can be used with multivectors to find new temperaments in an equivalent way to how new temperaments are found using concatenation with matrices. However, if the wedged multivectors are linearly dependent, then the multivector resulting from their wedge product will have all zeros for entries, and therefore it will not represent any temperament. Linear dependence does not impose this limitation on the concatenation of matrices approach, however; if equivalent linearly dependent matrices are concatenated, then a new matrix representing a new temperament will be produced. For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

=== Temperament addition ===

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 [[Basis|Bases]], such as [[Comma basis|comma bases]], are considered '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]]. In other words, that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.
+{{Wikipedia|Linear independence}}
 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.
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 === Wedge product ===
-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 merging|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. Ordinarily, the wedge product can be used with multivectors to find new temperaments in an equivalent way to how new temperaments are found using concatenation with matrices. However, if the wedged multivectors are linearly dependent, then the multivector resulting from their wedge product will have all zeros for entries, and therefore it will not represent any temperament. Linear dependence does not impose this limitation on the concatenation of matrices approach, however; if equivalent linearly dependent matrices are concatenated, then a new matrix representing a new temperament will be produced. For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].
 === Temperament addition ===