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In [[mathematics]], the '''(formal) complex conjugate''' of a [[complex numbers|complex]] [[vector space]] <math>V\,</math> is the complex vector space <math>\overline V</math> consisting of all formal [[complex conjugate]]s of elements of <math>V\,</math>.  That is, <math>\overline V</math> is a vector space whose elements are in [[bijection|one-to-one correspondence]] with the elements of <math>V\,</math>:
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
:<math>\overline V = \{\overline v \mid v \in V\},</math>
with the following rules for [[addition]] and [[scalar multiplication]]:
:<math>\overline v + \overline w = \overline{\,v+w\,}\quad\text{and}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}.</math>
Here <math>v\,</math> and <math>w\,</math> are vectors in <math>V\,</math>, <math>\alpha\,</math> is a complex number, and <math>\overline\alpha</math> denotes the complex conjugate of <math>\alpha\,</math>.


More concretely, the complex conjugate vector space is the same underlying ''real'' vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate [[linear complex structure]] ''J'' (different multiplication by ''i'').
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]]
* 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.


==Antilinear maps==
Registered users will be able to choose between the following three rendering modes:  
If <math>V\,</math> and <math>W\,</math> are complex vector spaces, a function <math>f\colon V \to W\,</math> is [[antilinear]] if
:<math>f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)</math>
for all <math>v,v'\in V\,</math> and <math>\alpha\in\mathbb{C}</math>.


One reason to consider the vector space <math>\overline V</math> is that it makes antilinear maps into [[linear map]]s.  Specifically, if <math>f\colon V \to W\,</math> is an antilinear map, then the corresponding map <math>\overline V \to W</math> defined by
'''MathML'''
:<math>\overline v \mapsto f(v)</math>
:<math forcemathmode="mathml">E=mc^2</math>
is linear.  Conversely, any linear map defined on <math>\overline V</math> gives rise to an antilinear map on <math>V\,</math>.


One way of thinking about this correspondence is that the map <math>C\colon V \to \overline V</math> defined by
<!--'''PNG'''  (currently default in production)
:<math>C(v) = \overline v</math>
:<math forcemathmode="png">E=mc^2</math>
is an antilinear bijection.  Thus if <math>f\colon \overline V \to W</math> is linear, then [[Function composition|composition]] <math>f \circ C\colon V \to W\,</math> is antilinear, and ''vice versa''.


==Conjugate linear maps==
'''source'''
Any linear map <math>f \colon V \to W\,</math> induces a '''conjugate linear map''' <math>\overline f \colon \overline V \to \overline W</math>, defined by the formula
:<math forcemathmode="source">E=mc^2</math> -->
:<math>\overline f (\overline v) = \overline{\,f(v)\,}.</math>
The conjugate linear map <math>\overline f</math> is linear.  Moreover, the [[identity function|identity map]] on <math>V\,</math> induces the identity map <math>\overline V</math>, and
:<math>\overline f \circ \overline g = \overline{\,f \circ g\,}</math>
for any two linear maps <math>f\,</math> and <math>g\,</math>.  Therefore, the rules <math>V\mapsto \overline V</math> and <math>f\mapsto\overline f</math> define a [[functor]] from the [[category theory|category]] of complex vector spaces to itself.


If <math>V\,</math> and  <math>W\,</math> are finite-dimensional and the map  <math>f\,</math> is described by the complex [[matrix (mathematics)|matrix]]  <math>A\,</math> with respect to the [[basis of a vector space|bases]]  <math>\mathcal B</math> of  <math>V\,</math> and  <math>\mathcal C</math> of  <math>W\,</math>, then the map  <math>\overline f</math> is described by the complex conjugate of  <math>A\,</math> with respect to the bases  <math>\overline{\mathcal B}</math> of  <math>\overline V</math> and  <math>\overline{\mathcal C}</math> of  <math>\overline W</math>.
<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].


==Structure of the conjugate==
==Demos==
The vector spaces <math>V\,</math> and <math>\overline V</math> have the same [[dimension of a vector space|dimension]] over the complex numbers and are therefore [[isomorphism|isomorphic]] as complex vector spaces. However, there is no [[natural isomorphism]] from  <math>V\,</math> to  <math>\overline V</math>.  (The map <math>C\,</math> is not an isomorphism, since it is antilinear.)


The double conjugate <math>\overline{\overline V}</math> is naturally isomorphic to <math>V\,</math>, with the isomorphism <math>\overline{\overline V} \to V</math> defined by
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
:<math>\overline{\overline v} \mapsto v.</math>
Usually the double conjugate of <math>V\,</math> is simply identified with <math>V\,</math>.


== Complex conjugate of a Hilbert space ==
Given a [[Hilbert space]] <math>\mathcal{H}</math> (either finite or infinite dimensional), its complex conjugate <math>\overline{\mathcal{H}}</math> is the same vector space as its [[continuous dual space]] <math>\mathcal{H}'</math>.
There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
In other words, any continuous [[linear functional]] on <math>\mathcal{H}</math> is an inner multiplication to some fixed vector, and vice versa.


Thus, the complex conjugate to a vector <math>v</math>, particularly in finite dimension case, may be denoted as <math>v^*</math> (v-star, a [[row vector]] which is the [[conjugate transpose]] to a column vector <math>v</math>).
* accessibility:
In quantum mechanics, the conjugate to a ''ket&nbsp;vector''&nbsp;<math>|\psi\rangle</math> is denoted as <math>\langle\psi|</math> – a ''bra vector'' (see [[bra–ket notation]]).
** 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.


==See also==
==Test pages ==
* [[Linear complex structure]]


==References==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Linear algebra]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
[[Category:Vectors|Vector space]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
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 .

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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