Gronwall's inequality - Revision history

en>Rjwilmsi: Journal cites, Added 1 doi to a journal cite using AWB (10365)

2014-08-13T14:23:04Z

Journal cites, Added 1 doi to a journal cite using AWB (10365)

en>Slawekb: Align is fixed now.

2014-02-09T11:37:00Z

Align is fixed now.

← Older revision		Revision as of 13:37, 9 February 2014
Line 1:		Line 1:
	~~{{expert\|mathematics\|date=October 2011}}~~		Another day I woke up and realized - At the moment I have been solitary for some time and after much bullying from buddies I now find myself signed-up for online dating. They promised me that there are lots of sweet, regular and interesting luke bryan on tour; [http://lukebryantickets.flicense.com simply click the following website page], people to fulfill, so here goes the message!<br>My family and friends are awesome and spending some time together at bar gigabytes or dishes is [https://www.Flickr.com/search/?q=consistently+vital consistently vital]. I haven't ever been in to cabarets as I realize that you could not [http://www.hotelsedinburgh.org luke bryan tour dates 2014] get a significant dialog against the noise. In addition, I have 2 definitely cheeky and very cute canines who are invariably eager to meet up new folks.<br>I attempt to stay as toned as potential being at the gymnasium many times weekly. I appreciate my athletics and attempt to play or see as several a potential. Being winter I am going [http://www.museodecarruajes.org concert luke bryan] to regularly at Hawthorn fits. Notice: In case that you considered buying an activity I do not mind, I have seen the carnage of fumbling fits at stocktake sales.<br><br>Look at my blog post ... luke bryan tickets klipsch; [http://lukebryantickets.pyhgy.com lukebryantickets.pyhgy.com],
	~~{{Unreferenced\|date=November 2009}}~~
	~~In [[mathematics]], specifically in [[linear algebra]],~~ the ~~'''coordinate space''', '''F'''<sup>''n''</sup>, is the prototypical example of an ''n''~~-~~dimensional [[vector space]] over a [[field (mathematics)\|field]] '''F'''~~. ~~It can be defined as the [[product space]]~~ of ~~F over a finite [[index set]].~~

	~~==Definition==~~
	~~Let '''F''' denote an arbitrary [[field (mathematics)\|field]] (such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''). For any [[Positive number\|positive]] [[integer]] ''n''~~, ~~the space of all ''n''-tuples of elements of '''F''' forms an ''n''-dimensional vector space over '''F''' called '''coordinate space'''~~ and ~~denoted '''F'''<sup>''n''</sup>.~~

	~~An element of '''F'''<sup>''n''</sup> is written~~
	:~~<math>\mathbf x = (x_1, x_2, \cdots, x_n)<~~/~~math>~~
	~~where each ''x''<sub>''i''<~~/~~sub> is an element of '''F'''~~. ~~The operations on '''F'''<sup>''n''</sup> are defined by~~
	~~:<math>\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)</math>~~
	~~:<math>\alpha \mathbf x = (\alpha x_1, \alpha x_2, \cdots, \alpha x_n)~~.~~</math>~~
	~~The zero vector is given by~~
	~~:<math>\mathbf 0 = (0~~, 0, ~~\cdots, 0)~~<~~/math~~>
	and ~~the additive inverse of the vector '''x'''~~ is ~~given by~~
	:~~<math>-\mathbf x = (-x_1, -x_2, \cdots, -x_n)~~.</~~math>~~

	~~===Matrix notation===~~
	~~In standard [[matrix (mathematics)\|matrix]] notation, each element of '''F'''<sup>''n''<~~/~~sup> is typically written as a [[column vector]]~~
	~~:<math>\mathbf x~~ = ~~\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>~~
	~~and sometimes as a [[row vector]~~]:
	~~:<math>\mathbf x = \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix}~~.~~</math>~~

	~~The coordinate space~~ '''F'''<sup>''n''</sup> may then be interpreted as the space of all ''n''×1 column vectors, or all 1×''n'' row vectors with the ordinary matrix operations of addition and scalar multiplication.

	~~[[Linear transformation]]s from '''F'''<sup>''n''</sup>~~ to ~~'''F'''<sup>''m''</sup> may then be written~~ as ''m''×''n'' matrices which act on the elements of '''F'''<sup>''n''</sup> via left multiplication (when the elements of '''F'''<sup>''n''</sup> are column vectors) or right multiplication (when they are row vectors).

	~~==Standard basis==~~
	~~The coordinate space '''F'''<sup>''n''</sup> comes with a [~~[~~standard basis]]:~~
	:~~<math>\mathbf e_1 = (1, 0, \ldots, 0)<~~/~~math>~~
	~~:<math>\mathbf e_2 = (0, 1, \ldots, 0)<~~/~~math>~~
	~~:<math>\vdots</math>~~
	~~:<math>\mathbf e_n = (0, 0, \ldots, 1)</math>~~
	~~where 1 denotes the multiplicative identity in '''F'''~~. ~~To see that this is~~ a ~~basis, note that an arbitrary vector in '''F'''<sup>''n''</sup> can be written uniquely in~~ the ~~form~~
	~~:<math>\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i~~.~~</math>~~

	~~==Discussion==~~
	~~It is a standard fact of linear algebra that every ''n''-dimensional vector space ''V'' over '''F''' is [[isomorphic]]~~ to ~~'''F'''<sup>''n''</sup>~~. ~~It is a crucial point, however, that this isomorphism is not [[canonical form\|canonical]]. If it were, mathematicians would work only with '''F'''~~<~~sup~~>~~''n''</sup> rather than with abstract vector spaces~~.

	~~A choice of isomorphism is equivalent~~ to ~~a choice of [[ordered basis]] for ''V''. To~~ see ~~this, let~~
	~~:''A'' : '''F'''<sup>''n''</sup> → ''V''~~
	be a ~~linear isomorphism~~. ~~Define an ordered basis {'''a'''<sub>''i''</sub>} for ''V'' by~~
	~~: '''a'''<sub>''i''</sub> = ''A''('''e'''<sub>''i''</sub>) for 1 ≤ ''i'' ≤ ''n''.~~
	~~Conversely, given any ordered basis {'''a'''<sub>''i''</sub>} for ''V'' define a linear map ''A''~~ : ~~'''F'''<sup>''n''<~~/~~sup> → ''V'' by~~
	~~:<math>A(\mathbf x) = \sum_{i=1}^n x_i \mathbf a_i.<~~/~~math>~~
	~~It is not hard to check that ''A'' is an isomorphism~~. ~~Thus ordered bases for ''V'' are in 1-1 correspondence with linear isomorphisms '''F'''<sup>''n''</sup> → ''V''~~.

	~~The reason for working with abstract vector spaces instead of '''F'''<sup>''n''</sup> is that it is often preferable~~ to ~~work in a ''coordinate-free'' manner, i~~.~~e. without choosing a preferred basis. Indeed, many vector spaces~~ that ~~naturally show up in mathematics~~ do not ~~come with a preferred choice~~ of ~~basis~~.

	~~It is possible and sometimes desirable to view a coordinate space [[duality (mathematics)\|dually]] as the set of F-valued functions on a finite set; that is, each "point" of '''F'''~~<~~sup~~>~~''n''~~<~~/sup~~> ~~is viewed as a function whose domain is the finite set {1,2~~....~~n} and codomain '''F'''. The function sends an element i of {1,2.~~...~~n} to the value of the i'th coordinate of the "point", so '''F'''<sup>''n''</sup> is, dually, a set of functions.~~

	~~==See also==~~
	*[[real coordinate space]], '''R'''<sup>''n''</sup>
	*[[complex coordinate space]], ~~'''C'''<sup>''n''</sup>~~
	*[[examples of vector spaces]]

	~~{{DEFAULTSORT:Coordinate Space}}~~
	~~[[Category:Linear algebra]]~~

en>Vkuncak: /* Proof */

2013-10-25T21:32:50Z

Proof

← Older revision		Revision as of 23:32, 25 October 2013
Line 1:		Line 1:
	<br><br>~~Another day I woke up and realized~~ - ~~Now I have also been solitary luke bryan tour dates for 2014~~ ([~~http://minioasis~~.~~com minioasis~~.~~com~~]~~) for some time~~ and ~~after much intimidation from buddies I today locate myself opted for online dating~~. ~~They promised me that there are lots~~ of ~~pleasant~~, ~~normal and fun individuals to meet~~, ~~therefore the pitch~~ is ~~gone~~ by ~~here!~~<br>~~My friends and family are wonderful and spending some [http~~://~~lukebryantickets~~.~~flicense.com photos luke bryan~~] ~~time together at pub gigs or dinners~~ is ~~definitely vital. I have never been in to dance~~ [~~http~~://~~www.hotelsedinburgh.org photos~~ of ~~luke bryan] clubs as I discover you could never have a good dialogue~~ with ~~all~~ the ~~noise~~. ~~I also have 2 unquestionably cheeky and quite cute puppies who are always eager~~ to ~~meet fresh people.~~<br>~~I try and keep~~ as ~~physically healthy as potential staying at~~ the ~~gymnasium many times~~ a ~~week~~. ~~I appreciate my sports and attempt to play or~~ see ~~since many~~ a ~~possible~~. ~~I shall regularly at Hawthorn fits being wintertime~~. ~~Notice: Supposing~~ that ~~you~~ would ~~contemplated shopping an athletics I don~~'~~t mind, I have observed the carnage~~ of ~~wrestling fits at~~ [~~http~~:/~~/pinterest~~.~~com~~/~~search~~/~~pins~~/~~?q=stocktake stocktake] revenue~~.<br><br>~~My [http~~://~~lukebryantickets~~.~~pyhgy~~.~~com garth brooks tour dates] webpage~~ ... [~~http:~~/~~/lukebryantickets~~.~~lazintechnologies~~.~~com~~ is ~~the luke bryan concert sold out~~]		{{expert\|mathematics\|date=October 2011}}
			{{Unreferenced\|date=November 2009}}
			In [[mathematics]], specifically in [[linear algebra]], the '''coordinate space''', '''F'''<sup>''n''</sup>, is the prototypical example of an ''n''-dimensional [[vector space]] over a [[field (mathematics)\|field]] '''F'''. It can be defined as the [[product space]] of F over a finite [[index set]].

			==Definition==
			Let '''F''' denote an arbitrary [[field (mathematics)\|field]] (such as the [[real number]]s '''R''' or the [[complex number]]s '''C'''). For any [[Positive number\|positive]] [[integer]] ''n'', the space of all ''n''-tuples of elements of '''F''' forms an ''n''-dimensional vector space over '''F''' called '''coordinate space''' and denoted '''F'''<sup>''n''</sup>.

			An element of '''F'''<sup>''n''</sup> is written
			:<math>\mathbf x = (x_1, x_2, \cdots, x_n)</math>
			where each ''x''<sub>''i''</sub> is an element of '''F'''. The operations on '''F'''<sup>''n''</sup> are defined by
			:<math>\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)</math>
			:<math>\alpha \mathbf x = (\alpha x_1, \alpha x_2, \cdots, \alpha x_n).</math>
			The zero vector is given by
			:<math>\mathbf 0 = (0, 0, \cdots, 0)</math>
			and the additive inverse of the vector '''x''' is given by
			:<math>-\mathbf x = (-x_1, -x_2, \cdots, -x_n).</math>

			===Matrix notation===
			In standard [[matrix (mathematics)\|matrix]] notation, each element of '''F'''<sup>''n''</sup> is typically written as a [[column vector]]
			:<math>\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>
			and sometimes as a [[row vector]]:
			:<math>\mathbf x = \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix}.</math>

			The coordinate space '''F'''<sup>''n''</sup> may then be interpreted as the space of all ''n''×1 column vectors, or all 1×''n'' row vectors with the ordinary matrix operations of addition and scalar multiplication.

			[[Linear transformation]]s from '''F'''<sup>''n''</sup> to '''F'''<sup>''m''</sup> may then be written as ''m''×''n'' matrices which act on the elements of '''F'''<sup>''n''</sup> via left multiplication (when the elements of '''F'''<sup>''n''</sup> are column vectors) or right multiplication (when they are row vectors).

			==Standard basis==
			The coordinate space '''F'''<sup>''n''</sup> comes with a [[standard basis]]:
			:<math>\mathbf e_1 = (1, 0, \ldots, 0)</math>
			:<math>\mathbf e_2 = (0, 1, \ldots, 0)</math>
			:<math>\vdots</math>
			:<math>\mathbf e_n = (0, 0, \ldots, 1)</math>
			where 1 denotes the multiplicative identity in '''F'''. To see that this is a basis, note that an arbitrary vector in '''F'''<sup>''n''</sup> can be written uniquely in the form
			:<math>\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>

			==Discussion==
			It is a standard fact of linear algebra that every ''n''-dimensional vector space ''V'' over '''F''' is [[isomorphic]] to '''F'''<sup>''n''</sup>. It is a crucial point, however, that this isomorphism is not [[canonical form\|canonical]]. If it were, mathematicians would work only with '''F'''<sup>''n''</sup> rather than with abstract vector spaces.

			A choice of isomorphism is equivalent to a choice of [[ordered basis]] for ''V''. To see this, let
			:''A'' : '''F'''<sup>''n''</sup> → ''V''
			be a linear isomorphism. Define an ordered basis {'''a'''<sub>''i''</sub>} for ''V'' by
			: '''a'''<sub>''i''</sub> = ''A''('''e'''<sub>''i''</sub>) for 1 ≤ ''i'' ≤ ''n''.
			Conversely, given any ordered basis {'''a'''<sub>''i''</sub>} for ''V'' define a linear map ''A'' : '''F'''<sup>''n''</sup> → ''V'' by
			:<math>A(\mathbf x) = \sum_{i=1}^n x_i \mathbf a_i.</math>
			It is not hard to check that ''A'' is an isomorphism. Thus ordered bases for ''V'' are in 1-1 correspondence with linear isomorphisms '''F'''<sup>''n''</sup> → ''V''.

			The reason for working with abstract vector spaces instead of '''F'''<sup>''n''</sup> is that it is often preferable to work in a ''coordinate-free'' manner, i.e. without choosing a preferred basis. Indeed, many vector spaces that naturally show up in mathematics do not come with a preferred choice of basis.

			It is possible and sometimes desirable to view a coordinate space [[duality (mathematics)\|dually]] as the set of F-valued functions on a finite set; that is, each "point" of '''F'''<sup>''n''</sup> is viewed as a function whose domain is the finite set {1,2....n} and codomain '''F'''. The function sends an element i of {1,2....n} to the value of the i'th coordinate of the "point", so '''F'''<sup>''n''</sup> is, dually, a set of functions.

			==See also==
			*[[real coordinate space]], '''R'''<sup>''n''</sup>
			*[[complex coordinate space]], '''C'''<sup>''n''</sup>
			*[[examples of vector spaces]]

			{{DEFAULTSORT:Coordinate Space}}
			[[Category:Linear algebra]]

en>ZéroBot: r2.7.1) (Robot: Adding ja:グロンウォールの不等式

2012-06-10T05:38:09Z

r2.7.1) (Robot: Adding ja:グロンウォールの不等式

New page

Another day I woke up and realized - Now I have also been solitary luke bryan tour dates for 2014 ([http://minioasis.com minioasis.com]) for some time and after much intimidation from buddies I today locate myself opted for online dating. They promised me that there are lots of pleasant, normal and fun individuals to meet, therefore the pitch is gone by here! My friends and family are wonderful and spending some [http://lukebryantickets.flicense.com photos luke bryan] time together at pub gigs or dinners is definitely vital. I have never been in to dance [http://www.hotelsedinburgh.org photos of luke bryan] clubs as I discover you could never have a good dialogue with all the noise. I also have 2 unquestionably cheeky and quite cute puppies who are always eager to meet fresh people. I try and keep as physically healthy as potential staying at the gymnasium many times a week. I appreciate my sports and attempt to play or see since many a possible. I shall regularly at Hawthorn fits being wintertime. Notice: Supposing that you would contemplated shopping an athletics I don't mind, I have observed the carnage of wrestling fits at [http://pinterest.com/search/pins/?q=stocktake stocktake] revenue. My [http://lukebryantickets.pyhgy.com garth brooks tour dates] webpage ... [http://lukebryantickets.lazintechnologies.com is the luke bryan concert sold out]