Quotient space (linear algebra)
Encyclopedia
In linear algebra
, the quotient of a vector space
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).
over a field
K, and let N be a subspace
of V. We define an equivalence relation
~ on V by stating that x ~ y if x − y ∈ N. 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 equivalent to the zero vector; in other words all the vectors in N get mapped into the equivalence class of the zero vector.
The equivalence class of x is often denoted
since it is given by
The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by
It is not hard to check that these operations are well-defined
(i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].
The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.
Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such that only the first m entries are non-zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner.
More generally, if V is an (internal) direct sum of subspaces U and W:
then the quotient space V/U is naturally isomorphic to W .
from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel
(or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence
If U is a subspace of V, the dimension
of V/U is called the codimension
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 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, it follows that the codimension of U in V is the difference between the dimensions of V and U :
Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x ∈ V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem
: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
The cokernel
of a linear operator T : V → W is defined to be the quotient space W/im(T).
and M is a closed
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
The quotient space X/M is complete
with respect to the norm, so it is a Banach space.
If X is a Hilbert space
, then the quotient space X/M is isomorphic to the orthogonal complement of M.
Then X/M is a locally convex space, and the topology on it is the quotient topology.
If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space
, then so is X/M .
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the quotient of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V by a subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
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 spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, and let N be a subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of V. We define an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ on V by stating that x ~ y if x − y ∈ N. 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 equivalent to the zero vector; in other words all the vectors in N get mapped into the equivalence class of the zero vector.
The equivalence class of x is often denoted
- [x] = x + N
since it is given by
- [x] = {x + n : n ∈ N}.
The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by
- α[x] = [αx] for all α ∈ K, and
- [x] + [y] = [x+y].
It is not hard to check that these operations are well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
(i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].
The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.
Examples
Let X = R2 be the standard Cartesian plane, and let Y be a line 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. This gives one way in which to visualize quotient spaces geometrically.Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such that only the first m entries are non-zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner.
More generally, if V is an (internal) direct sum of subspaces U and W:
then the quotient space V/U is naturally isomorphic to W .
Properties
There is a natural epimorphismEpimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
(or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence
If U is a subspace of V, the dimension
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
of V/U is called the codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
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 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, it follows that the codimension of U in V is the difference between the dimensions of V and U :
Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x ∈ V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem
Rank-nullity theorem
In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...
: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
The cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
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 spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
and M is a closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
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
The quotient space X/M is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
with respect to the norm, so it is a Banach space.
Examples
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.If X is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, then the quotient space X/M is isomorphic to the orthogonal complement of M.
Generalization to locally convex spaces
The quotient of a locally convex space by a closed subspace is again locally convex . Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα|α∈A} where A is an index set. Let M be a closed subspace, and define seminorms qα by on X/MThen X/M is a locally convex space, and the topology on it is the quotient topology.
If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space
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...
, then so is X/M .
See also
- quotient set
- quotient groupQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- quotient moduleQuotient moduleIn abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...
- quotient spaceQuotient spaceIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
(in topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
)