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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'').
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


== Definition ==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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; in other words all the vectors in ''N'' get mapped into the equivalence class of the zero vector.
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The [[equivalence class]] of ''x'' is often denoted
Registered users will be able to choose between the following three rendering modes:  
:[''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
'''MathML'''
*α[''x''] = [α''x''] for all α ∈ ''K'', and
:<math forcemathmode="mathml">E=mc^2</math>
*[''x'']&nbsp;+&nbsp;[''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''&nbsp;&isin;&nbsp;''V'' the equivalence class [''v''] is known as the '''quotient map'''.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


== Examples ==
'''source'''
Let ''X''&nbsp;=&nbsp;'''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.
:<math forcemathmode="source">E=mc^2</math> -->


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 only the first ''m'' entries are non-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''&minus;''m'' coordinates. The quotient space  '''R'''<sup>''n''</sup>/ '''R'''<sup>''m''</sup> is [[isomorphic]] to  '''R'''<sup>''n''&minus;''m''</sup> in an obvious manner.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


More generally, if ''V'' is an (internal) [[direct sum of vector spaces|direct sum]] of subspaces ''U'' and ''W'':
==Demos==
:<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]].
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


== 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]]
* accessibility:
:<math>0\to U\to V\to V/U\to 0.\,</math>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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}}:
==Test pages ==
:<math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math>


Let ''T'' : ''V'' &rarr; ''W'' be a [[linear operator]]. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' &isin; ''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'').
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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The [[cokernel]] of a linear operator ''T'' : ''V'' &rarr; ''W'' is defined to be the quotient space ''W''/im(''T'').
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*[[Url2Image|Url2Image (private Wikis only)]]
== Quotient of a Banach space by a subspace ==
==Bug reporting==
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
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<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'' &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 ''C''[0,1]&nbsp;/&nbsp;''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>&alpha;</sub>|&alpha;&isin;''A''} where ''A'' is an index set.  Let ''M'' be a closed subspace, and define seminorms ''q''<sub>&alpha;</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]] (in [[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]]
 
[[ca:Espai vectorial quocient]]
[[de:Faktorraum]]
[[it:Spazio vettoriale quoziente]]
[[he:מרחב מנה (אלגברה לינארית)]]
[[ja:商線型空間]]
[[pl:Przestrzeń ilorazowa (algebra liniowa)]]
[[ru:Факторпространство по подпространству]]
[[zh:商空间 (线性代数)]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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