Linear dependence: Difference between revisions

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Basis ~~vector sets~~ are '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]], ~~which means~~ that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.

[[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.

By a ''basis'' vector set — such as a [[comma basis]] — it is meant that the set does not simply only contain the vectors that have been explicitly listed in order to represent the set, but the set actually contains the entire infinitude of vectors that are possible to form through linear combinations of the explicit vectors. For example, the comma basis {{bra|{{vector|4 -4 1}}}} does not only have a single member, {{vector|4 -4 1}}. It also contains {{vector|8 -8 2}}, {{vector|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector|-4 4 -1}}. And the comma basis {{bra|{{vector|4 -4 1}} {{vector|7 0 -3}}}} doesn't merely include {{vector|4 -4 1}} and {{vector|7 0 -3}}; it also includes {{vector|4 -4 1}} + {{vector|7 0 -3}} = {{vector|11 -4 -2}}, and 2·{{vector|4 -4 1}} + -1·{{vector|7 0 -3}} = {{vector|1 -8 5}}, and many many more. In [[regular temperament theory]], this mathematical property corresponds to the fact that if you temper out a set of commas, then you've also effectively tempered out any interval that's equal to combinations of those commas. So when we say that ''basis'' vector sets share ''basis'' vectors, we mean that even if the shared vector is not immediately explicit, it can be found within each basis ~~vector set in this way.~~

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~~ 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 arithmetic]], which are defined for objects that can be interpreted as basis vector sets, such as matrices or multivectors, and that also represent regular temperaments.

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== Linear dependence as defined for various types of vector sets ==

Linear dependence is defined for several objects relevant to RTT that can be defined as vector sets. These objects will each be discussed in detail below.

Linear dependence is defined for several objects relevant to RTT that can be defined as basis vector sets. These objects will each be discussed in detail below.

=== Linear dependence between matrices ===

=== Linear dependence between basis matrices ===

Linear dependence is defined on sets of matrices, such as two temperaments' mappings, or two temperaments' comma bases.

Linear dependence is defined on sets of basis matrices (matrices acting as bases), such as two temperaments' mappings<ref>Mappings are not typically thought of as bases, but their row vectors can be considered to span rowspaces in an analogous way that comma bases span spaces.</ref>, or two temperaments' comma bases.

A set of matrices are linear dependent upon each other when some vector can be found where each matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{ket|{{map|5 8 12}} {{map|7 11 16}}}} and {{ket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{ket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.

A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{ket|{{map|5 8 12}} {{map|7 11 16}}}} and {{ket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{ket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.

Sometimes matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis {{bra|{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis {{bra|{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:

Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis {{bra|{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis {{bra|{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:

* {{vector|4 -4 1 0}} = {{vector|-26 15 1 0}} - {{vector|-30 19 0 0}}

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* {{vector|13 -10 0 1}} = {{vector|-6 2 0 1}} - {{vector|-19 12 0 0}}

These two matrices are the comma bases dual to the 7-limit [[uniform map]]s for 12-ET and 19-ET, respectively. {{vector|4 -4 1 0}} is the meantone comma and {{vector|13 -10 0 1}} is Harrison's comma, so we can say that both of these temperaments temper out both of these commas.

These two basis matrices are the comma bases dual to the 7-limit [[uniform map]]s for 12-ET and 19-ET, respectively. {{vector|4 -4 1 0}} is the meantone comma and {{vector|13 -10 0 1}} is Harrison's comma, so we can say that both of these temperaments temper out both of these commas.

==== For a given set of matrices, how to compute a basis for their linearly dependent vectors ====

==== For a given set of basis matrices, how to compute a basis for their linearly dependent vectors ====

A basis for the linearly dependent vectors of a set of matrices can be computed using the operations [[meet and join]].

A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear dependence basis <math>L_{\text{dep}}</math> can be computed using the operations [[meet and join]].

* To check if two mappings are linearly dependent, we use a meet. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives a basis ~~for their linearly dependent vectors~~.

* To check if two mappings are linearly dependent, we use a meet. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear dependence basis.

* To check if two comma bases are linearly dependent, we use a join. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a basis ~~for their linearly dependent vectors~~.

* To check if two comma bases are linearly dependent, we use a join. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear dependence basis.

Certainly there are other ways to determine linear dependency, but this method is handy because if the matrices ''are'' linearly dependent, then it also gives you ~~a basis for the vectors by which they are linearly dependent~~.

Certainly there are other ways to determine linear dependency, but this method is handy because if the basis matrices ''are'' linearly dependent, then it also gives you <math>L_{\text{dep}}</math>.

=== Linear dependence between multivectors ===

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Linear dependence is defined for sets of multivectors, such as two temperaments' multimaps, or two temperaments' multicommas. For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linear dependence between multivectors]].

=== Linear dependence within a single matrix ===

=== Linear dependence within a single basis matrix ===

Linear dependence is defined among the basis vectors of a single matrix. For more information, see [[rank-deficient|rank-deficiency and full-rank]].

Linear dependence is defined among the basis vectors of a single basis matrix. For more information, see [[rank-deficient|rank-deficiency and full-rank]].

=== Linear dependence between individual vectors ===

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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 matrices and performing a meet or join. 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 meet or join operation from 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 meet or join. 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 meet or join operation from 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 arithmetic ===

@@ Line 1: / Line 1: @@
-Basis vector sets are '''linearly dependent''' when they share a common basis [[Wikipedia:Vector|vector]], which means that they can form an identical vector through [[Wikipedia:Linear_combinations|linear combinations]] of their member vectors.
+[[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.
-By a ''basis'' vector set — such as a [[comma basis]] — it is meant that the set does not simply only contain the vectors that have been explicitly listed in order to represent the set, but the set actually contains the entire infinitude of vectors that are possible to form through linear combinations of the explicit vectors. For example, the comma basis {{bra|{{vector|4 -4 1}}}} does not only have a single member, {{vector|4 -4 1}}. It also contains {{vector|8 -8 2}}, {{vector|12 -12 3}}, and all possible multiples of this vector, including negative ones like {{vector|-4 4 -1}}. And the comma basis {{bra|{{vector|4 -4 1}} {{vector|7 0 -3}}}} doesn't merely include {{vector|4 -4 1}} and {{vector|7 0 -3}}; it also includes {{vector|4 -4 1}} + {{vector|7 0 -3}} = {{vector|11 -4 -2}}, and 2·{{vector|4 -4 1}} + -1·{{vector|7 0 -3}} = {{vector|1 -8 5}}, and many many more. In [[regular temperament theory]], this mathematical property corresponds to the fact that if you temper out a set of commas, then you've also effectively tempered out any interval that's equal to combinations of those commas. So when we say that ''basis'' vector sets share ''basis'' vectors, we mean that even if the shared vector is not immediately explicit, it can be found within each basis vector set in this way.
+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 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 arithmetic]], which are defined for objects that can be interpreted as basis vector sets, such as matrices or multivectors, and that also represent regular temperaments.
@@ Line 9: / Line 7: @@
 == Linear dependence as defined for various types of vector sets ==
-Linear dependence is defined for several objects relevant to RTT that can be defined as vector sets. These objects will each be discussed in detail below.
+Linear dependence is defined for several objects relevant to RTT that can be defined as basis vector sets. These objects will each be discussed in detail below.
-=== Linear dependence between matrices ===
+=== Linear dependence between basis matrices ===
-Linear dependence is defined on sets of matrices, such as two temperaments' mappings, or two temperaments' comma bases.
+Linear dependence is defined on sets of basis matrices (matrices acting as bases), such as two temperaments' mappings<ref>Mappings are not typically thought of as bases, but their row vectors can be considered to span rowspaces in an analogous way that comma bases span spaces.</ref>, or two temperaments' comma bases.
-A set of matrices are linear dependent upon each other when some vector can be found where each matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{ket|{{map|5 8 12}} {{map|7 11 16}}}} and {{ket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{ket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.
+A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{ket|{{map|5 8 12}} {{map|7 11 16}}}} and {{ket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{ket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.
-Sometimes matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis {{bra|{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis {{bra|{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:
+Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis {{bra|{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis {{bra|{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:
 * {{vector|4 -4 1 0}} = {{vector|-26 15 1 0}} - {{vector|-30 19 0 0}}
@@ Line 25: / Line 23: @@
 * {{vector|13 -10 0 1}} = {{vector|-6 2 0 1}} - {{vector|-19 12 0 0}}
-These two matrices are the comma bases dual to the 7-limit [[uniform map]]s for 12-ET and 19-ET, respectively. {{vector|4 -4 1 0}} is the  meantone comma and {{vector|13 -10 0 1}} is Harrison's comma, so we can say that both of these temperaments temper out both of these commas.
+These two basis matrices are the comma bases dual to the 7-limit [[uniform map]]s for 12-ET and 19-ET, respectively. {{vector|4 -4 1 0}} is the  meantone comma and {{vector|13 -10 0 1}} is Harrison's comma, so we can say that both of these temperaments temper out both of these commas.
-==== For a given set of matrices, how to compute a basis for their linearly dependent vectors ====
+==== For a given set of basis matrices, how to compute a basis for their linearly dependent vectors ====
-A basis for the linearly dependent vectors of a set of matrices can be computed using the operations [[meet and join]].
+A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear dependence basis <math>L_{\text{dep}}</math> can be computed using the operations [[meet and join]].
-* To check if two mappings are linearly dependent, we use a meet. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives a basis for their linearly dependent vectors.
+* To check if two mappings are linearly dependent, we use a meet. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear dependence basis.
-* To check if two comma bases are linearly dependent, we use a join. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a basis for their linearly dependent vectors.
+* To check if two comma bases are linearly dependent, we use a join. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear dependence basis.
-Certainly there are other ways to determine linear dependency, but this method is handy because if the matrices ''are'' linearly dependent, then it also gives you a basis for the vectors by which they are linearly dependent.
+Certainly there are other ways to determine linear dependency, but this method is handy because if the basis matrices ''are'' linearly dependent, then it also gives you <math>L_{\text{dep}}</math>.
 === Linear dependence between multivectors ===
@@ Line 39: / Line 37: @@
 Linear dependence is defined for sets of multivectors, such as two temperaments' multimaps, or two temperaments' multicommas. For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linear dependence between multivectors]].
-=== Linear dependence within a single matrix ===
+=== Linear dependence within a single basis matrix ===
-Linear dependence is defined among the basis vectors of a single matrix. For more information, see [[rank-deficient|rank-deficiency and full-rank]].
+Linear dependence is defined among the basis vectors of a single basis matrix. For more information, see [[rank-deficient|rank-deficiency and full-rank]].
 === Linear dependence between individual vectors ===
@@ Line 55: / Line 53: @@
 === 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 matrices and performing a meet or join. 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 meet or join operation from 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 meet or join. 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 meet or join operation from 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 arithmetic ===