Linear dependence: Difference between revisions

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== Variance ==

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 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.

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 == Variance ==
-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 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.