quotient vector space
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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, the quotient of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' by a subspace ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a quotient space and is denoted ''V''/''N'' (read "''V'' mod ''N''" or "''V'' by ''N''").


Definition

Formally, the construction is as follows. Let ''V'' be a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field ''K'', and let ''N'' be a subspace of ''V''. We define an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
~ on ''V'' by stating that ''x'' ~ ''y'' if . That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. From this definition, one can deduce that any element of ''N'' is related to the zero vector; more precisely, all the vectors in ''N'' get mapped into the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the zero vector. The equivalence class – or, in this case, the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
– of ''x'' is often denoted : 'x''= ''x'' + ''N'' since it is given by : 'x''= . The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes induced by ~ on ''V''. Scalar multiplication and addition are defined on the equivalence classes by *α 'x''= ±''x''for all α ∈ ''K'', and * 'x''nbsp;+  'y''= 'x'' + ''y'' It is not hard to check that these operations are
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
(i.e. do not depend on the choice of representatives). These operations turn the quotient space ''V''/''N'' into a vector space over ''K'' with ''N'' being the zero class, The mapping that associates to the equivalence class 'v''is known as the quotient map. Alternatively phrased, the quotient space V/N is the set of all affine subsets of V which are
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
to


Examples


Lines in Cartesian Plane

Let be the standard
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, and let ''Y'' be a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
through the origin in ''X''. Then the quotient space ''X''/''Y'' can be identified with the space of all lines in ''X'' which are parallel to ''Y''. That is to say that, the elements of the set ''X''/''Y'' are lines in ''X'' parallel to ''Y''. Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to ''Y''. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to ''Y''. Similarly, the quotient space for R3 by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.)


Subspaces of Cartesian Space

Another example is the quotient of R''n'' by the subspace spanned by the first ''m'' standard basis vectors. The space R''n'' consists of all ''n''-tuples of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s . The subspace, identified with R''m'', consists of all ''n''-tuples such that the last ''n'' − ''m'' entries are zero: . Two vectors of R''n'' are in the same equivalence class modulo the subspace
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
they are identical in the last ''n'' − ''m'' coordinates. The quotient space R''n''/R''m'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to R''n''−''m'' in an obvious manner.


Polynomial Vector Space

Let \mathcal_3(\mathbb) be the vector space of all cubic polynomials over the real numbers. Then \mathcal_3(\mathbb) / \langle x^2 \rangle is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is \, while another element of the quotient space is \.


General Subspaces

More generally, if ''V'' is an (internal)
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of subspaces ''U'' and ''W,'' :V=U\oplus W then the quotient space ''V''/''U'' is
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to ''W''.


Lebesgue Integrals

An important example of a functional quotient space is an L''p'' space.


Properties

There is a natural
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class 'x'' The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
(or nullspace) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
:0\to U\to V\to V/U\to 0.\, If ''U'' is a subspace of ''V'', the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of ''V''/''U'' is called the
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
of ''U'' in ''V''. Since a basis of ''V'' may be constructed from a basis ''A'' of ''U'' and a basis ''B'' of ''V''/''U'' by adding a
representative Representative may refer to: Politics * Representative democracy, type of democracy in which elected officials represent a group of people * House of Representatives, legislative body in various countries or sub-national entities * Legislator, som ...
of each element of ''B'' to ''A'', the dimension of ''V'' is the sum of the dimensions of ''U'' and ''V''/''U''. If ''V'' is
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
, it follows that the codimension of ''U'' in ''V'' is the difference between the dimensions of ''V'' and ''U'': :\mathrm(U) = \dim(V/U) = \dim(V) - \dim(U). Let ''T'' : ''V'' → ''W'' be a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' in ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
for vector spaces says that the quotient space ''V''/ker(''T'') is isomorphic to the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of ''V'' in ''W''. An immediate
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
, for finite-dimensional spaces, is the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
: the dimension of ''V'' is equal to the dimension of the kernel (the nullity of ''T'') plus the dimension of the image (the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of ''T''). The
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of a linear operator ''T'' : ''V'' → ''W'' is defined to be the quotient space ''W''/im(''T'').


Quotient of a Banach space by a subspace

If ''X'' is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
and ''M'' is a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on ''X''/''M'' by : \, \, _ = \inf_ \, x-m\, _X = \inf_ \, x+m\, _X = \inf_\, y\, _X. When ''X'' is complete, then the quotient space ''X''/''M'' is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
with respect to the norm, and therefore a Banach space.


Examples

Let ''C'' ,1denote the Banach space of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
real-valued functions on the interval ,1with the
sup norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the ...
. Denote the subspace of all functions ''f'' ∈ ''C'' ,1with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space is isomorphic to R. If ''X'' is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, then the quotient space ''X''/''M'' is isomorphic to the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of ''M''.


Generalization to locally convex spaces

The quotient of a
locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
by a closed subspace is again locally convex. Indeed, suppose that ''X'' is locally convex so that the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on ''X'' is generated by a family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s where ''A'' is an index set. Let ''M'' be a closed subspace, and define seminorms ''q''α on ''X''/''M'' by :q_\alpha( = \inf_ p_\alpha(v). Then ''X''/''M'' is a locally convex space, and the topology on it is the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
. If, furthermore, ''X'' is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
, then so is ''X''/''M''. If ''X'' is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
, then so is ''X''/''M''. p. 54, § 12.11.3


See also

*
Quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
*
Quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
*
Quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
*
Quotient space (topology) In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...


References


Sources

* * * * * {{Linear algebra Functional analysis Linear algebra Space (linear algebra)