Linear dependence - Revision history

FloraC: +Wikipedia box

2024-06-25T08:24:31Z

+Wikipedia box

← Older revision		Revision as of 08:24, 25 June 2024
Line 1:		Line 1:
	[[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.		[[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.		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.

Cmloegcmluin: /* Wedge product */

2024-06-24T20:06:35Z

Wedge product

← Older revision		Revision as of 20:06, 24 June 2024
Line 53:		Line 53:
	=== Wedge product ===		=== 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 ===		=== Temperament addition ===

Cmloegcmluin: /* Why row-rank always equals column-rank */ eliminate unnecessary use of antitranspose; make same point more simply with a single transpose

2024-02-11T02:17:00Z

Why row-rank always equals column-rank: eliminate unnecessary use of antitranspose; make same point more simply with a single transpose

← Older revision		Revision as of 02:17, 11 February 2024
Line 121:		Line 121:

	<nowiki>		<nowiki>
	~~reverseEachRow[a_] := Reverse[a, 2];~~
	~~reverseEachCol[a_] := Reverse[a];~~
	~~antitranspose[a_] := reverseEachRow[reverseEachCol[a]];~~

	P = {		P = {
	{1, 4/3, 4/3},		{ 1, 4/3, 4/3 },
	{0, -4/3, -1/3},		{ 0, -4/3, -1/3 },
	{0, 4/3, 1/3}		{ 0, 4/3, 1/3 }
	};		};
	Last[HermiteDecomposition[P]]		Last[HermiteDecomposition[P]]
	~~antitranspose[~~Last[HermiteDecomposition[~~antitranspose~~[P]]]]		Last[HermiteDecomposition[Transpose[P]]

	→ {		→ {
	{ 1, 0, 1},		{ 1, 0, 1 },
	{ 0, 4/3, 1/3},		{ 0, 4/3, 1/3 },
	{ 0, 0, 0}		{ 0, 0, 0 }
	}		}
	→ {		→ {
	{ 0, 0, 0},		{ 1/3, 2/3, -2/3 },
	{-1, ~~4, 0~~},		{ 0, 1, -1 },
	{ 0, ~~4/3~~, ~~1/3~~}		{ 0, 0, 0 }
	} </nowiki>		} </nowiki>

	On the other hand, we have the [[minimax-E-copfr-S]] (or primes miniRMS-U) tuning of 12-ET, where <math>d</math> still equals <math>3</math> but now <math>r = 1</math>:		On the other hand, we have the [[minimax-E-copfr-S]] (or primes miniRMS-U) tuning of 12-ET, where <math>d</math> still equals <math>3</math> but now <math>r = 1</math>:

	<nowiki>M = {{12, 19, 28}};		<nowiki>M = {{ 12, 19, 28 }};
	G = PseudoInverse[M]		G = PseudoInverse[M]
	→ {{12, 19, 28}} / 1289		→ {{ 12, 19, 28 }} / 1289

	P = G.M		P = G.M
	→ {		→ {
	{ 12² , 12·19, 12·28},		{ 12² , 12·19, 12·28 },
	{19·12, 19² , 19·28},		{ 19·12, 19² , 19·28 },
	{28·12, 28·19, 28² }		{ 28·12, 28·19, 28² }
	} / 1289		} / 1289

Line 160:		Line 156:
	antitranspose[Last[HermiteDecomposition[antitranspose[P]]]]		antitranspose[Last[HermiteDecomposition[antitranspose[P]]]]
	→ {		→ {
	{12, 19, 28},		{ 12, 19, 28 },
	{ 0, 0, 0},		{ 0, 0, 0 },
	{ 0, 0, 0}		{ 0, 0, 0 }
	}/1289		}/1289
	→ {		→ {
	{ 0, 0, 0},		{ 12, 19, 28 },
	{ 0, 0, 0},		{ 0, 0, 0 },
	{12, 19, 28}		{ 0, 0, 0 }
	}/1289 </nowiki>		}/1289 </nowiki>

Cmloegcmluin: /* Rank-deficiency and full-rank */ add diagrams

2022-12-31T19:17:30Z

Rank-deficiency and full-rank: add diagrams

← Older revision		Revision as of 19:17, 31 December 2022
Line 70:		Line 70:

	==Rank-deficiency and full-rank==		==Rank-deficiency and full-rank==
			[[File:Full-rank - updated.png\|thumb\|600x600px\|Rank of a matrix can be revealed by reducing a matrix such as with Hermite normal form (row or column style). The shapes of the reduced matrices are shown in green within the original matrix, and the all-zero rows and columns in red. ]]
	A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)\|rank]] equals whichever is smaller between its width (column count) and height (row count):		A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)\|rank]] equals whichever is smaller between its width (column count) and height (row count):
	* For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.		* For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.
Line 107:		Line 107:

	===Why row-rank always equals column-rank===		===Why row-rank always equals column-rank===
			[[File:Row-rank is col-rank.png\|left\|thumb\|600x600px\|Here we see an example matrix 𝐴 expressed as the product of matrices 𝑋 and 𝑌. These two matrices' shared dimension, 𝑟, is both the row-rank and the column-rank of 𝐴. In the middle and bottom rows of this diagram, we see how 𝐴 can be described in two different ways: in the middle row we see each of its rows as a linear combination of the 𝑟 rows of 𝑌, and in the bottom row we see each of its columns as a linear combination of the 𝑟 columns of 𝑋. So whether we reduce it row-style (HNF) or column-style (CHNF), we find the rank to be 𝑟. ]]
	Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.		Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.

Cmloegcmluin: more information about full-rank and rank-deficiency

2022-12-31T19:06:30Z

more information about full-rank and rank-deficiency

Show changes

Cmloegcmluin: add link to concept of vanish / temper out

2022-12-27T18:11:55Z

add link to concept of vanish / temper out

← Older revision		Revision as of 18:11, 27 December 2022
Line 23:		Line 23:
	* {{vector\|13 -10 0 1}} = {{vector\|-6 2 0 1}} - {{vector\|-19 12 0 0}}		* {{vector\|13 -10 0 1}} = {{vector\|-6 2 0 1}} - {{vector\|-19 12 0 0}}

	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.		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 make both of these commas [[vanish]].

	==== For a given set of basis 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 ====

Cmloegcmluin: EBK updates

2022-12-27T16:48:17Z

EBK updates

← Older revision		Revision as of 16:48, 27 December 2022
Line 13:		Line 13:
	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.		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 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}}.		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 {{rket\|{{map\|5 8 12}} {{map\|7 11 16}}}} and {{rket\|{{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 {{rket\|{{map\|1 0 -4}} {{map\|0 1 4}}}} and {{rket\|{{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 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}}:		Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis [{{vector\|-30 19 0 0}} {{vector\|-26 15 1 0}} {{vector\|-17 9 0 1}}] and the comma basis [{{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}}		* {{vector\|4 -4 1 0}} = {{vector\|-26 15 1 0}} - {{vector\|-30 19 0 0}}

Cmloegcmluin: link to temperament merging

2022-01-16T00:32:21Z

link to temperament merging

← Older revision		Revision as of 00:32, 16 January 2022
Line 28:		Line 28:

	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 [[temperament merging]].		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 [[temperament merging]].
	* To check if two mappings are linearly dependent, we use a comma-merge. 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 mappings are linearly dependent, we use a [[comma-merging\|comma-merge]]. 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 map-merge. 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.		* To check if two comma bases are linearly dependent, we use a [[map-merging\|map-merge]]. 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 basis matrices ''are'' linearly dependent, then it also gives you <math>L_{\text{dep}}</math>.		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>.
Line 53:		Line 53:
	=== Wedge product ===		=== 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 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 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]]

	=== Temperament addition ===		=== Temperament addition ===

Cmloegcmluin: consistent hyphenation of "prime-count vector"

2022-01-12T19:56:50Z

consistent hyphenation of "prime-count vector"

← Older revision		Revision as of 19:56, 12 January 2022
Line 61:		Line 61:
	== Variance ==		== 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.		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.

Cmloegcmluin: temperament arithmetic → temperament addition

2022-01-11T07:34:03Z

temperament arithmetic → temperament addition

← Older revision		Revision as of 07:34, 11 January 2022
Line 3:		Line 3:
	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 ~~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.		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 ==
Line 47:		Line 47:
	=== Linear dependence between temperaments ===		=== Linear dependence between temperaments ===

	The conditions of temperament ~~arithmetic~~ motivate a special definition of linear dependence for temperaments. For more information, see: [[Temperament ~~arithmetic~~#2. Linear dependence between temperaments]].		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 ==
Line 55:		Line 55:
	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 ~~arithmetic~~ ===		=== Temperament addition ===

	Temperament ~~arithmetic~~ 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 ~~arithmetic~~#Addability]].		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 ==
Line 63:		Line 63:
	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 ~~arithmetic~~#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.		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 ==