en>Mark viking: /* Tensor manipulation */ Removed peacock term 'very'

2013-09-17T20:42:39Z

Tensor manipulation: Removed peacock term 'very'

← Older revision		Revision as of 22:42, 17 September 2013
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	~~Hello~~, ~~my name~~ is ~~Andrew~~ and ~~my spouse doesn~~'~~t like~~ it ~~at all~~. It's not a ~~common factor but what I like doing~~ is to ~~climb but I don~~'t ~~have the time lately~~. ~~Distributing production~~ is ~~exactly where her primary income arrives from~~. ~~For years he~~'~~s been living in Mississippi and he doesn~~'~~t strategy on altering it~~.<br><br>~~Feel free~~ to ~~visit my web page ::~~ [~~http:~~//~~cartoonkorea~~.~~com~~/~~ce002~~/~~1093612 free psychic~~]		In [[mathematics]], a '''dependence relation''' is a [[binary relation]] which generalizes the relation of [[linear dependence]].

			Let <math>X</math> be a [[set (mathematics)\|set]]. A (binary) relation <math>\triangleleft</math> between an element <math>a</math> of <math>X</math> and a [[subset]] <math>S</math> of <math>X</math> is called a ''dependence relation'', written <math>a \triangleleft S</math>, if it satisfies the following properties:
			* if <math>a \in S</math>, then <math>a \triangleleft S</math>;
			* if <math>a \triangleleft S</math>, then there is a [[finite set\|finite]] subset <math>S_0</math> of <math>S</math>, such that <math>a \triangleleft S_0</math>;
			* if <math>T</math> is a subset of <math>X</math> such that <math>b \in S</math> implies <math>b \triangleleft T</math>, then <math>a \triangleleft S</math> implies <math>a \triangleleft T</math>;
			* if <math>a \triangleleft S</math> but <math>a \not\!\triangleleft S-\lbrace b \rbrace</math> for some <math>b \in S</math>, then <math>b \triangleleft (S-\lbrace b \rbrace)\cup\lbrace a \rbrace</math>.

			Given a ''dependence relation'' <math>\triangleleft</math> on <math>X</math>, a subset <math>S</math> of <math>X</math> is said to be ''independent'' if <math>a \not\!\triangleleft S - \lbrace a \rbrace</math> for all <math>a \in S.</math> If <math>S \subseteq T</math>, then <math>S</math> is said to ''span'' <math>T</math> if <math>t \triangleleft S</math> for every <math>t \in T.</math> <math>S</math> is said to be a ''basis'' of <math>X</math> if <math>S</math> is ''independent'' and <math>S</math> ''spans'' <math>X.</math>

			'''Remark.''' If <math>X</math> is a non-empty set with a dependence relation <math>\triangleleft</math>, then <math>X</math> always has a basis with respect to <math>\triangleleft.</math> Furthermore, any two bases of <math>X</math> have the same [[cardinality]].

			==Examples==
			* Let <math>V</math> be a [[vector space]] over a [[field (mathematics)\|field]] <math>F.</math> The relation <math>\triangleleft</math>, defined by <math>\upsilon \triangleleft S</math> if <math>\upsilon</math> is in the [[linear subspace\|subspace]] spanned by <math>S</math>, is a dependence relation. This is [[logical equivalence\|equivalent]] to the definition of [[linear independence\|linear dependence]].
			* Let <math>K</math> be a [[field extension]] of <math>F.</math> Define <math>\triangleleft</math> by <math>\alpha \triangleleft S</math> if <math>\alpha</math> is [[algebraic element\|algebraic]] over <math>F(S).</math> Then <math>\triangleleft</math> is a dependence relation. This is equivalent to the definition of [[algebraic dependence]].

			==See also==
			* [[matroid]]

			{{PlanetMath attribution\|id=5792\|title=Dependence relation}}

			[[Category:Linear algebra]]
			[[Category:Mathematical relations]]

70.247.173.254: tweak

2012-07-22T23:55:27Z

tweak

New page

Hello, my name is Andrew and my spouse doesn't like it at all. It's not a common factor but what I like doing is to climb but I don't have the time lately. Distributing production is exactly where her primary income arrives from. For years he's been living in Mississippi and he doesn't strategy on altering it.<br><br>Feel free to visit my web page :: [http://cartoonkorea.com/ce002/1093612 free psychic]

Penrose graphical notation - Revision history

en>Mark viking: /* Tensor manipulation */ Removed peacock term 'very'

70.247.173.254: tweak