Editing Quotient space (linear algebra)

{{Short description|Vector space consisting of affine subsets}}
{{about|quotients of vector spaces|quotients of topological spaces|Quotient space (topology)}}

In [[linear algebra]], the '''quotient''' of a [[vector space]] <math>V</math> by a [[linear subspace|subspace]] <math>N</math> is a vector space obtained by "collapsing" <math>N</math> to zero. The space obtained is called a '''quotient space''' and is denoted <math>V/N</math> (read "<math>V</math> mod <math>N</math>" or "<math>V</math> by <math>N</math>").

== Definition ==
Formally, the construction is as follows.<ref>{{Harvard citation text|Halmos|1974}} pp. 33-34 §§ 21-22</ref> Let <math>V</math> be a [[vector space]] over a [[field (mathematics)|field]] <math>\mathbb{K}</math>, and let <math>N</math> be a [[linear subspace|subspace]] of <math>V</math>. We define an [[equivalence relation]] <math>\sim</math> on <math>V</math> by stating that <math>x \sim y</math> iff {{nowrap| <math>x - y \in N</math>}}. That is, <math>x</math> is related to <math>y</math> if and only if one can be obtained from the other by adding an element of <math>N</math>. This definition implies that any element of <math>N</math> is related to the zero vector; more precisely, all the vectors in <math>N</math> get mapped into the [[equivalence class]] of the zero vector.

The equivalence class – or, in this case, the [[coset]] – of <math>x</math> is defined as 
:<math>[x] := \{ x + n: n \in N \}</math>
and is often denoted using the shorthand <math>[x] = x + N</math>.

The quotient space <math>V/N</math> is then defined as <math>V/_\sim</math>, the set of all equivalence classes induced by <math>\sim</math> on <math>V</math>. Scalar multiplication and addition are defined on the equivalence classes by<ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 9 § 1.2.4</ref><ref>{{Harvard citation text|Roman|2005}} p. 75-76, ch. 3</ref> 
*<math>\alpha [x] = [\alpha x]</math> for all <math>\alpha \in \mathbb{K}</math>, and
*<math>[x] + [y] = [x+y]</math>.
It is not hard to check that these operations are [[well-defined]] (i.e. do not depend on the choice of [[representative (mathematics)|representatives]]). These operations turn the quotient space <math>V/N</math> into a vector space over <math>\mathbb{K}</math> with <math>N</math> being the zero class, <math>[0]</math>.

The mapping that associates to {{nowrap|<math>v \in V</math>}} the equivalence class <math>[v]</math> is known as the '''quotient map'''.

Alternatively phrased, the quotient space <math>V/N</math> is the set of all [[Affine space|affine subsets]] of <math>V</math> which are [[Parallel (geometry)|parallel]] to {{nowrap|<math>N</math>.}}<ref>{{Harvard citation text|Axler|2015}} p. 95, § 3.83</ref>

== Examples ==

===Lines in Cartesian Plane===

Let {{nowrap|1=''X'' = '''R'''<sup>2</sup>}} be the standard [[Cartesian plane]], and let ''Y'' be a [[line (geometry)|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''.  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 '''R'''<sup>3</sup> 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 (geometry)|plane]] which only intersects the line at the origin.)

===Subspaces of Cartesian Space===

Another example is the quotient of '''R'''<sup>''n''</sup> by the subspace spanned by the first ''m'' [[standard basis vector]]s.  The space '''R'''<sup>''n''</sup> consists of all ''n''-tuples of [[real number]]s {{nowrap|(''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: {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''m''</sub>, 0, 0, ..., 0)}}.  Two vectors of '''R'''<sup>''n''</sup> are in the same equivalence 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.

===Polynomial Vector Space===

Let <math>\mathcal{P}_3(\mathbb{R})</math> be the vector space of all cubic polynomials over the real numbers. Then <math>\mathcal{P}_3(\mathbb{R}) / \langle x^2 \rangle </math> 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 <math>\{x^3 + a x^2 - 2x + 3 : a \in \mathbb{R}\}</math>, while another element of the quotient space is <math>\{a x^2 + 2.7 x : a \in \mathbb{R}\}</math>.

===General Subspaces===

More generally, if ''V'' is an (internal) [[direct sum]] of subspaces ''U'' and ''W,''
:<math>V=U\oplus W</math>
then the quotient space ''V''/''U'' is [[Natural transformation|naturally isomorphic]] to ''W''.<ref>{{Harvard citation text|Halmos|1974}} p. 34, § 22, Theorem 1</ref>

===Lebesgue Integrals===

An important example of a functional quotient space is an [[Lp space#Lp spaces and Lebesgue integrals|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 (linear 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 (linear algebra)|basis]] of ''V'' may be constructed from a basis ''A'' of ''U'' and a basis ''B'' of ''V''/''U'' by adding a [[representative (mathematics)|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 [[dimension (vector space)|finite-dimensional]], it follows that the codimension of ''U'' in ''V'' is the difference between the dimensions of ''V'' and ''U'':<ref>{{Harvard citation text|Axler|2015}} p. 97, § 3.89</ref><ref>{{Harvard citation text|Halmos|1974}} p. 34, § 22, Theorem 2</ref>
:<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'' in ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The [[first isomorphism theorem]] for vector spaces says that the quotient space ''V''/ker(''T'') is isomorphic to the [[image (mathematics)|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 [[kernel (linear algebra)|nullity]] of ''T'') plus the dimension of the image (the [[rank (linear algebra)|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 (mathematics)|norm]] on ''X''/''M'' by
:<math> \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X = \inf_{m \in M} \|x+m\|_X = \inf_{y\in [x]}\|y\|_X. </math>

=== Examples ===
Let ''C''[0,1] denote the Banach space of [[continuous function|continuous]] real-valued [[function (mathematics)|functions]] on the [[interval (mathematics)|interval]] [0,1] with the [[sup norm]]. Denote the subspace of all functions ''f'' &isin; ''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 {{nowrap|''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.<ref>{{Harvard citation text|Dieudonné|1976}} p. 65, § 12.14.8</ref>  Indeed, suppose that ''X'' is locally convex so that the [[topological space|topology]] on ''X'' is generated by a family of [[seminorm]]s {''p''<sub>α</sub>&nbsp;|&nbsp;α&nbsp;&isin;&nbsp;''A''} where ''A'' is an index set.  Let ''M'' be a closed subspace, and define seminorms ''q''<sub>α</sub> on ''X''/''M'' by

:<math>q_\alpha([x]) = \inf_{v\in [x]} p_\alpha(v).</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''.<ref>{{Harvard citation text|Dieudonné|1976}} p. 54, § 12.11.3</ref>

==See also==
* [[Quotient group]]
* [[Quotient module]]
* [[Quotient set]]
* [[Quotient space (topology)]]

== References ==
<references />

==Sources==
* {{Cite book|last=Axler|first=Sheldon|title=Linear Algebra Done Right|publisher=[[Springer Science+Business Media | Springer]]|year=2015|isbn=978-3-319-11079-0|edition=3rd|series=[[Undergraduate Texts in Mathematics]]|location=|pages=|author-link=Sheldon Axler}}
*{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=[[Treatise on Analysis]]|publisher=[[Academic Press]]|year=1976|volume=2|pages=|isbn=978-0122155024}}
*{{Cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher=[[Springer Science+Business Media | Springer]]|year=1974|isbn=0-387-90093-4|edition=2nd|series=[[Undergraduate Texts in Mathematics]]|volume=|location=|pages=|author-link=Paul Halmos|orig-year=1958}}
*{{Cite book|last=Katznelson|first=Yitzhak|title=A (Terse) Introduction to Linear Algebra|last2=Katznelson|first2=Yonatan R.|publisher=[[American Mathematical Society]]|isbn=978-0-8218-4419-9|volume=|publication-date=2008|pages=|author-link=Yitzhak Katznelson}}
*{{Cite book|last=Roman|first=Steven|title=Advanced Linear Algebra|publisher=[[Springer Science+Business Media|Springer]]|year=2005|isbn=0-387-24766-1|edition=2nd|series=[[Graduate Texts in Mathematics]]|location=|pages=|author-link=Steven Roman}}

{{Linear algebra}}

[[Category:Functional analysis]]
[[Category:Linear algebra]]
[[Category:Quotient objects|Space (linear algebra)]]