Zobel network - Revision history

en>Gadget850: refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB

2014-12-31T14:15:28Z

refbegin, replaced: </div> → {{refend}}, <div class="references-small"> → {{refbegin}} using AWB

← Older revision		Revision as of 16:15, 31 December 2014
Line 1:		Line 1:
	~~In [[mathematics]], a '''reality structure'''~~ on ~~a [[complex vector space]] ''V'' is a decomposition~~ of ~~''V'' into two real subspaces, called~~ the ~~[[real part\|real]]~~ and [~~[imaginary part]]s of ''V''~~:		Hi there. My title is Sophia Meagher even though it is not the name on my beginning certification. One of the issues she loves most is canoeing and she's been doing it for quite a while. Credit authorising accurate psychic readings ([http://help.ksu.edu.sa/node/65129 http://help.ksu.edu.sa/node/65129]) is how she makes a residing. Alaska is the only location I've been residing in but [http://www.chk.woobi.co.kr/xe/?document_srl=346069 telephone psychic] now I'm considering other options.<br><br>Take a look at my web blog - [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ tarot card readings]
	:~~<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.<~~/~~math>~~
	~~Here ''V''<sub>'''R'''<~~/~~sub> is a real subspace of ''V'', i~~.e. ~~a subspace of ''V'' considered as a [[vector space]] over the [[real number]]s~~. ~~If ''V'' has [[complex dimension]~~] ~~''n'' (real dimension 2''n''~~)~~, then ''V''<sub>'''R'''</sub> must have real dimension ''n''~~.

	~~The '''standard reality structure''' on the vector space <math>\mathbb{C}^n</math>~~ is the ~~decomposition~~
	~~:<math>\mathbb{C}^n = \mathbb{R}^n \oplus i\,\mathbb{R}^n.</math>~~

	~~In the presence of a reality structure, every vector in '~~'~~V'' has a real part and an imaginary part, each of which is a vector~~ in ~~''V''<sub>'''R'''</sub>:~~
	~~:<math>v = \operatorname{Re}\{v\}+i\,\operatorname{Im}\{v\}</math>~~
	~~In this case, the [~~[~~complex conjugate]] of a vector ''v'' is defined as follows~~:
	~~:<math>\overline v = \operatorname{Re}\{v\} - i\,\operatorname{Im}\{v\}<~~/~~math>~~
	~~This map <math>v \mapsto \overline v<~~/~~math> is an [[antilinear]] [[Involution (mathematics)\|involution]], i~~.e.
	~~:<math>\overline{\overline v} = v,\quad \overline{v + w} = \overline{v} + \overline{w},\quad\text{and}\quad~~
	~~\overline{\alpha v} = \overline\alpha \, \overline{v}~~.</~~math>~~

	~~Conversely, given an antilinear involution <math>v \mapsto c(v)<~~/~~math> on a complex vector space~~ '~~'V'', it is possible to define a reality structure on ''V'' as follows~~. ~~Let~~
	:<~~math~~>~~\operatorname{Re}\{v\}=\frac{1}{2}\left(v + c(v)\right),~~<~~/math~~>
	~~and define~~
	:~~<math>V_\mathbb{R} = \left\{\operatorname{Re}\{v\} \mid v \in V \right\}~~.</~~math>~~
	~~Then~~
	~~:<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.<~~/~~math>~~
	~~This is actually the decomposition of ''V'' as the [[eigenspace]]s of the real [[linear operator]] ''c''. The eigenvalues of ''c'' are +1 and −1, with eigenspaces ''V''<sub>'''R'''<~~/~~sub> and <math>i<~~/~~math> ''V''<sub>'''R'''<~~/~~sub>, respectively. Typically, the operator ''c'' itself, rather than the eigenspace decomposition it entails, is referred to as the '''reality structure''' on ''V''.~~

	~~==See also==~~
	*[[Linear complex structure]]
	*[[Complexification]]

	~~==References==~~
	* {{Citation \| last1=Penrose \| first1=Roger \| author1-~~link=Roger Penrose \| last2=Rindler \| first2=Wolfgang \| author2~~-~~link=Wolfgang Rindler \| title=Spinors and space-time. Vol. 2 \| publisher=[[Cambridge University Press]] \| series=Cambridge Monographs on Mathematical Physics \| isbn=978~~-0-~~521~~-~~25267~~-~~6 \| id={{MathSciNet \| id = 838301}} \| year=1986}}~~

	~~[[Category:Linear algebra]~~]


	~~{{Linear-algebra-stub}}~~

en>Omnipaedista: per WP:HNP

2013-10-31T16:25:59Z

per WP:HNP

← Older revision		Revision as of 18:25, 31 October 2013
Line 1:		Line 1:
	~~I would like to introduce myself to you~~, ~~I am Andrew~~ and ~~my spirit messages (~~[~~http~~://~~kard~~.dk/?p=~~24252 http://kard~~.dk/~~]) spouse doesn~~'~~t like it at all. For years she~~'~~s been operating as~~ a ~~journey agent. What me~~ and ~~my family adore~~ is ~~bungee leaping but I~~'~~ve been taking~~ [~~http~~://~~www~~.~~youronlinepublishers~~.~~com~~/~~authWiki~~/~~AdolphvhBladenqq phone psychic]~~ on ~~new things lately. Mississippi is where his home~~ is.<br><br>~~Also visit my blog post~~ ... ~~spirit messages ([http:~~//si.~~dgmensa~~.~~org/xe/index.php?document_srl~~=~~48014&mid~~=~~c0102 http://si~~.~~dgmensa~~.~~org/xe/index.php?document_srl~~=~~48014&mid~~=~~c0102~~])		In [[mathematics]], a '''reality structure''' on a [[complex vector space]] ''V'' is a decomposition of ''V'' into two real subspaces, called the [[real part\|real]] and [[imaginary part]]s of ''V'':
			:<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.</math>
			Here ''V''<sub>'''R'''</sub> is a real subspace of ''V'', i.e. a subspace of ''V'' considered as a [[vector space]] over the [[real number]]s. If ''V'' has [[complex dimension]] ''n'' (real dimension 2''n''), then ''V''<sub>'''R'''</sub> must have real dimension ''n''.

			The '''standard reality structure''' on the vector space <math>\mathbb{C}^n</math> is the decomposition
			:<math>\mathbb{C}^n = \mathbb{R}^n \oplus i\,\mathbb{R}^n.</math>

			In the presence of a reality structure, every vector in ''V'' has a real part and an imaginary part, each of which is a vector in ''V''<sub>'''R'''</sub>:
			:<math>v = \operatorname{Re}\{v\}+i\,\operatorname{Im}\{v\}</math>
			In this case, the [[complex conjugate]] of a vector ''v'' is defined as follows:
			:<math>\overline v = \operatorname{Re}\{v\} - i\,\operatorname{Im}\{v\}</math>
			This map <math>v \mapsto \overline v</math> is an [[antilinear]] [[Involution (mathematics)\|involution]], i.e.
			:<math>\overline{\overline v} = v,\quad \overline{v + w} = \overline{v} + \overline{w},\quad\text{and}\quad
			\overline{\alpha v} = \overline\alpha \, \overline{v}.</math>

			Conversely, given an antilinear involution <math>v \mapsto c(v)</math> on a complex vector space ''V'', it is possible to define a reality structure on ''V'' as follows. Let
			:<math>\operatorname{Re}\{v\}=\frac{1}{2}\left(v + c(v)\right),</math>
			and define
			:<math>V_\mathbb{R} = \left\{\operatorname{Re}\{v\} \mid v \in V \right\}.</math>
			Then
			:<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.</math>
			This is actually the decomposition of ''V'' as the [[eigenspace]]s of the real [[linear operator]] ''c''. The eigenvalues of ''c'' are +1 and −1, with eigenspaces ''V''<sub>'''R'''</sub> and <math>i</math> ''V''<sub>'''R'''</sub>, respectively. Typically, the operator ''c'' itself, rather than the eigenspace decomposition it entails, is referred to as the '''reality structure''' on ''V''.

			==See also==
			*[[Linear complex structure]]
			*[[Complexification]]

			==References==
			* {{Citation \| last1=Penrose \| first1=Roger \| author1-link=Roger Penrose \| last2=Rindler \| first2=Wolfgang \| author2-link=Wolfgang Rindler \| title=Spinors and space-time. Vol. 2 \| publisher=[[Cambridge University Press]] \| series=Cambridge Monographs on Mathematical Physics \| isbn=978-0-521-25267-6 \| id={{MathSciNet \| id = 838301}} \| year=1986}}

			[[Category:Linear algebra]]


			{{Linear-algebra-stub}}

en>Khazar2: /* Half sections */clean up, typos fixed: half way → halfway using AWB (8097)

2012-08-25T18:11:10Z

Half sections: clean up, typos fixed: half way → halfway using AWB (8097)

New page

I would like to introduce myself to you, I am Andrew and my spirit messages ([http://kard.dk/?p=24252 http://kard.dk/]) spouse doesn't like it at all. For years she's been operating as a journey agent. What me and my family adore is bungee leaping but I've been taking [http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq phone psychic] on new things lately. Mississippi is where his home is.<br><br>Also visit my blog post ... spirit messages ([http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102])

Zobel network - Revision history

en>Gadget850: refbegin, replaced: → {{refend}}, → {{refbegin}} using AWB

en>Omnipaedista: per WP:HNP

en>Khazar2: /* Half sections */clean up, typos fixed: half way → halfway using AWB (8097)

en>Gadget850: refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB