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In [[linear algebra]], the '''quotient''' of a [[vector space]] ''V'' by a [[linear subspace|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 {{harv|Halmos|1974|loc=§21-22}}. Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K'', and let ''N'' be a [[linear subspace|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 related to the zero vector; more precisely 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''] = [α''x''] for all α ∈ ''K'', and | |||
*[''x''] + [''y''] = [''x''+''y'']. | |||
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'''. | |||
== Examples == | |||
Let ''X'' = '''R'''<sup>2</sup> 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 '''R'''<sup>''n''</sup> by the subspace spanned by the first ''m'' standard basis vectors. The space '''R'''<sup>''n''</sup> consists of all ''n''-tuples of real numbers (''x''<sub>1</sub>,…,''x''<sub>''n''</sub>). The subspace, identified with '''R'''<sup>''m''</sup>, consists of all ''n''-tuples such that the last ''n-m'' entries are zero: (''x''<sub>1</sub>,…,''x''<sub>''m''</sub>,0,0,…,0). Two vectors of '''R'''<sup>''n''</sup> 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 '''R'''<sup>''n''</sup>/ '''R'''<sup>''m''</sup> is [[isomorphic]] to '''R'''<sup>''n''−''m''</sup> in an obvious manner. | |||
More generally, if ''V'' is an (internal) [[direct sum of vector spaces|direct sum]] of subspaces ''U'' and ''W,'' | |||
:<math>V=U\oplus W</math> | |||
then the quotient space ''V''/''U'' is naturally isomorphic to ''W'' {{harv|Halmos|1974|loc=Theorem 22.1}}. | |||
An important example of a functional quotient space is a [[Lp_space#Lp_spaces|L<sup>p</sup> space]]. | |||
== Properties == | |||
There is a natural [[epimorphism]] from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The [[kernel (algebra)|kernel]] (or [[nullspace]]) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the [[short exact sequence]] | |||
:<math>0\to U\to V\to V/U\to 0.\,</math> | |||
If ''U'' is a subspace of ''V'', the [[dimension (vector space)|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'' {{harv|Halmos|1974|loc=Theorem 22.2}}: | |||
:<math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math> | |||
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''). | |||
== Quotient of a Banach space by a subspace == | |||
If ''X'' is a [[Banach space]] and ''M'' is a [[closed set|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 | |||
:<math> \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X. </math> | |||
The quotient space ''X''/''M'' is [[complete space|complete]] 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]], then the quotient space ''X''/''M'' is isomorphic to the [[Hilbert space#Orthogonal complements and projections|orthogonal complement]] of ''M''. | |||
=== Generalization to locally convex spaces === | |||
The quotient of a [[locally convex space]] by a closed subspace is again locally convex {{harv|Dieudonné|1970|loc=12.14.8}}. Indeed, suppose that ''X'' is locally convex so that the topology on ''X'' is generated by a family of [[seminorm]]s {''p''<sub>α</sub>|α∈''A''} where ''A'' is an index set. Let ''M'' be a closed subspace, and define seminorms ''q''<sub>α</sub> by on ''X''/''M'' | |||
:<math>q_\alpha([x]) = \inf_{x\in [x]} p_\alpha(x).</math> | |||
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'' {{harv|Dieudonné|1970|loc=12.11.3}}. | |||
==See also== | |||
*[[quotient set]] | |||
*[[quotient group]] | |||
*[[quotient module]] | |||
*[[quotient space (topology)]] | |||
==References== | |||
* {{citation|first=Paul|last=Halmos|authorlink=Paul Halmos|title=Finite dimensional vector spaces|publisher=Springer|year=1974|isbn=978-0-387-90093-3}}. | |||
* {{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on analysis, Volume II|publisher=Academic Press|year=1970}}. | |||
[[Category:Linear algebra]] | |||
[[Category:Functional analysis]] | |||
Revision as of 09:59, 28 February 2013
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).
Definition
Formally, the construction is as follows Template:Harv. Let V be a vector space 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 related to the zero vector; more precisely 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 (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 the last n-m entries are 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 Template:Harv.
An important example of a functional quotient space is a Lp space.
Properties
There is a natural epimorphism 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 Template:Harv:
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).
Quotient of a Banach space by a subspace
If X is a Banach space 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.
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, 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 Template:Harv. 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/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 Template:Harv.
See also
References
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