Linear dependence

Vector sets are collinear, or linearly dependent, when they share a common vector, meaning that they can form an identical vector through linear combinations of their member vectors.

When vector sets do not share a common vector in this way, they are noncollinear, or linearly independent. Collinear vector sets are in a sense more closely related to each other than noncollinear vector sets.

Collinearity is involved in certain operations used in regular temperament theory, such as the wedge product or temperament arithmetic, which are defined for objects that can be interpreted as vector sets, such as matrices or multivectors, and that also represent regular temperaments.

Collinearity as defined for various types of vector sets

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

Matrix collinearity

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

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

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

• [4 -4 1 0⟩ = [-26 15 1 0⟩ - [-30 19 0 0⟩
• [4 -4 1 0⟩ = [-15 8 1 0⟩ - [-19 12 0 0⟩

• [13 -10 0 1⟩ = [-17 9 0 1⟩ - [-30 19 0 0⟩
• [13 -10 0 1⟩ = [-6 2 0 1⟩ - [-19 12 0 0⟩

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

How to compute collinearity of matrices

Matrix collinearity can be computed using the operations meet and join.

• To check if two mappings are collinear, 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 noncollinear, and otherwise the mappings are collinear and the result gives their shared vectors.
• To check if two comma bases are collinear, we use a join. This process exactly parallels the process for checking two mappings for collinearity. 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 noncollinear, and otherwise they are collinear and the result gives their shared vectors.

Multivector collinearity

Collinearity 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#Collinearity of multivectors.

Intra-matrix collinearity

Collinearity is defined among vectors of a single matrix. For more information, see rank-deficiency and full-rank.

Individual vector collinearity

Collinearity is defined between sets where each contains only a single vector. The sense in which individual vectors like this can be collinear is the simplest of all: it is only if they are multiples of each other. For example, ⟨12 19 28] and ⟨24 38 56] are collinear, because ⟨24 38 56] = 2⟨12 19 28]. But ⟨12 19 28] and ⟨12 19 27] are not.

Temperament collinearity

The conditions of temperament arithmetic motivate a special definition of collinearity for temperaments. For more information, see: Temperament arithmetic#2. Temperament collinearity.

RTT applications involving collinearity

Wedge product

Collinearity has an interesting effect on the wedge product, which otherwise produces the same result on vectors that one finds by treating these vectors as matrices and performing a meet or join. The wedge product of any two collinear multivectors, their wedge product will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for noncollinear 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#Collinearity exception

Temperament arithmetic

Temperament arithmetic only results in a usable temperament when the input temperaments are monononcollinear, an advanced property that builds upon collinearity. For more information, see Temperament arithmetic#Monononcollinearity.

Variance

Collinearity is defined both for vector sets whether they are covariant ("covectors", such as maps) or contravariant (plain "vectors", such as prime count vectors). For simplicity, this article will use the word "vector" in its general sense, which includes either plain/contravariant vectors or (covariant) covectors.

Plain vectors and covectors cannot be compared with each other, however. Collinearity is only defined for a set of vector sets, or a set of covector sets. Collinearity is not defined for a set including both vector sets and covector sets. For example, a set including one mapping (a covector set) and one comma basis (a plain vector set) has no directly meaningful notion of collinearity. 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 geometry

The "linear" part of the name "collinear" comes from the fact that it is defined by linear combinations of vectors. This notion of collinearity comes from linear algebra, a field whose name uses the term "linear" in the same underlying sense: of a linear equation (that is, an equation with no variables raised to a power higher than 1).

This linear algebra notion of collinearity is different from but related to the notion of collinearity in geometry, where three points found on the same line are said to be collinear. In geometry and linear algebra, the simplest objects are both represented by a single coordinate: in geometry this object is a point, and in linear algebra it is a vector. The critical difference between these two objects is that geometric points are zero-dimensional, simply representing a position in space, whereas linear algebra vectors are one-dimensional, representing both a magnitude and direction. Vectors manage to encode this extra dimension without providing any additional information because they are understood to describe a position in space relative to an origin.

In geometrical terms, a vector could be considered to be a directed line segment.

When used in geometry, collinearity often treats the root "linear" more literally, referring only to sets of (zero-dimensional) points that are all found along the same (one-dimensional) line. For higher-dimensional cases, such as sets of (one-dimensional) lines that are all found on the same (two-dimensional) plane, or sets of (two-dimensional) planes that are all found in the same (three-dimensional) volume, specialized terms like "coplanar" or "covolumetric"/"cohyperplanar" may be used to differentiate from "collinear". However, even in geometry, sometimes "collinear" is still used generically for any of these cases.

When used in linear algebra, however, where all objects are essentially described by lines (or, more accurately, directed line segments), collinearity is always generic to the dimensionality, referring to any dimensionality of objects that share such lines.