12-240/Classnotes for Tuesday October 09 - Revision history

Peterlue at 10:06, 7 December 2012

2012-12-07T10:06:28Z

← Older revision		Revision as of 06:06, 7 December 2012
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	2/ It is linearly independent		2/ It is linearly independent

	== ~~theorems~~ ==		== Theorems ==
	1/ β is a basis of V iff every element of V can be written as a linear combination of elements of β in a unique way.		1/ β is a basis of V iff every element of V can be written as a linear combination of elements of β in a unique way.

Peterlue: /* Clarification on lecture notes */

2012-12-07T10:02:52Z

Clarification on lecture notes

← Older revision		Revision as of 06:02, 7 December 2012
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	<b>Theorem 1.5:</b> The span of any subset <math>S</math> of a vector space <math>V</math> is a subspace of <math>V</math>. Moreover, any subspace of <math>V</math> that contains <math>S</math> must also contain <math>span(S)</math>		<b>Theorem 1.5:</b> The span of any subset <math>S</math> of a vector space <math>V</math> is a subspace of <math>V</math>. Moreover, any subspace of <math>V</math> that contains <math>S</math> must also contain <math>span(S)</math>

	Since <math>\beta</math> is a subset of <math>V</math>, <math>span(\beta)</math> is a subspace of <math>V</math> from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that <math>G \subseteq span(\beta)</math>. From the "Moreover" part of Theorem 1.5, since <math>span(\beta)</math> is a subspace of <math>V</math> containing <math>G</math>, <math>span(\beta)</math> must also contain <math>span(G)</math>.		Since <math>\beta</math> is a subset of <math>V</math>, <math>span(\beta)</math> is a subspace of <math>V</math> from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that <math>G \subseteq span(\beta)</math>. From the "Moreover" part of Theorem 1.5, since <math> span(\beta)</math> is a subspace of <math>V</math> containing <math>G</math>, <math> span(\beta)</math> must also contain <math> span(G)</math>.

	== Lecture notes scanned by [[User:Oguzhancan\|Oguzhancan]] ==		== Lecture notes scanned by [[User:Oguzhancan\|Oguzhancan]] ==

Peterlue: /* theorems */

2012-12-07T09:59:44Z

theorems

← Older revision		Revision as of 05:59, 7 December 2012
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	Assume <math>\sum \!\,</math> ai∙ui = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β		Assume <math>\sum \!\,</math> ai∙ui = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β

	<math>\sum \!\,</math> ai∙ui = 0 = <math>\sum \!\,</math> ~~0.ui~~		<math>\sum \!\,</math> ai∙ui = 0 = <math>\sum \!\,</math> 0∙ui

	since 0 can be written as a linear combination of elements of β in a unique way, ai=0 <math>\forall\!\,</math> i		since 0 can be written as a linear combination of elements of β in a unique way, ai=0 <math>\forall\!\,</math> i

Peterlue: /* theorems */

2012-12-07T09:59:15Z

theorems

← Older revision		Revision as of 05:59, 7 December 2012
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	The fact that β span is the fact that every element of V can be written as a linear combination of elements of β, which is given		The fact that β span is the fact that every element of V can be written as a linear combination of elements of β, which is given

	Assume <math>\sum \!\,</math> ~~ai.ui~~ = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β		Assume <math>\sum \!\,</math> ai∙ui = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β

	<math>\sum \!\,</math> ~~ai.ui~~ = 0 = <math>\sum \!\,</math> 0.ui		<math>\sum \!\,</math> ai∙ui = 0 = <math>\sum \!\,</math> 0.ui

	since 0 can be written as a linear combination of elements of β in a unique way, ai=0 <math>\forall\!\,</math> i		since 0 can be written as a linear combination of elements of β in a unique way, ai=0 <math>\forall\!\,</math> i

Peterlue: /* Definition of basis */

2012-12-07T09:57:30Z

Definition of basis

← Older revision		Revision as of 05:57, 7 December 2012
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	In this lecture, the professor concentrated on bases and related theorems.		In this lecture, the professor concentrated on bases and related theorems.
	== Definition of basis ==		== Definition of basis ==
	β <math>\subset \!\,</math> V is a ~~basic~~ if		β <math>\subset \!\,</math> V is a basis if

	1/ It generates ( span) V, span β = V		1/ It generates (span) V, span β = V

	2/ It is linearly independent		2/ It is linearly independent

Peterlue at 09:56, 7 December 2012

2012-12-07T09:56:54Z

← Older revision		Revision as of 05:56, 7 December 2012
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	{{12-240/Navigation}}		{{12-240/Navigation}}
	In this lecture, the professor ~~concentrate~~ on ~~basics~~ and related theorems.		In this lecture, the professor concentrated on bases and related theorems.
	== Definition of basis ==		== Definition of basis ==
	β <math>\subset \!\,</math> V is a basic if		β <math>\subset \!\,</math> V is a basic if

Peterlue: /* theorems */

2012-12-07T09:55:22Z

theorems

← Older revision		Revision as of 05:55, 7 December 2012
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	== theorems ==		== theorems ==
	1/ β is a ~~basic~~ of V iff every element of V can be written as a linear combination of elements of β in a unique way.		1/ β is a basis of V iff every element of V can be written as a linear combination of elements of β in a unique way.

	proof: ( in the case β is finite)		proof: ( in the case β is finite)

Peterlue: /* Definition of basic */

2012-12-07T09:54:58Z

Definition of basic

← Older revision		Revision as of 05:54, 7 December 2012
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	{{12-240/Navigation}}		{{12-240/Navigation}}
	In this lecture, the professor concentrate on basics and related theorems.		In this lecture, the professor concentrate on basics and related theorems.
	== Definition of ~~basic~~ ==		== Definition of basis ==
	β <math>\subset \!\,</math> V is a basic if		β <math>\subset \!\,</math> V is a basic if

Peterlue: /* theorems */

2012-12-07T09:49:09Z

theorems

← Older revision		Revision as of 05:49, 7 December 2012
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	(=>) every element of V can be written as a linear combination of elements of β in a unique way.		(=>) every element of V can be written as a linear combination of elements of β in a unique way.

	So, suppose <math>\sum \!\,</math> ~~ai.ui~~ = v = <math>\sum \!\,</math> ~~bi.ui~~		So, suppose <math>\sum \!\,</math> ai∙ui = v = <math>\sum \!\,</math> bi∙ui

	Thus <math>\sum \!\,</math> ~~ai.ui~~ - <math>\sum \!\,</math> ~~bi.ui~~ = 0		Thus <math>\sum \!\,</math> ai∙ui - <math>\sum \!\,</math> bi∙ui = 0

	<math>\sum \!\,</math> (ai-bi)~~.ui~~ = 0		<math>\sum \!\,</math> (ai-bi)∙ui = 0

	β is linear independent hence (ai - bi)= 0 <math>\forall\!\,</math> i		β is linear independent hence (ai - bi)= 0 <math>\forall\!\,</math> i

Grace.zhu: /* Lecture notes scanned by gracez */

2012-11-04T18:14:08Z

Lecture notes scanned by gracez

← Older revision		Revision as of 14:14, 4 November 2012
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	</gallery>		</gallery>

	== Lecture notes ~~scanned~~ by [[User:Grace.zhu\|gracez]] ==		== Lecture notes uploaded by [[User:Grace.zhu\|gracez]] ==
	<gallery>		<gallery>
	Image:12-240-O9-1.jpg\|Page 1		Image:12-240-O9-1.jpg\|Page 1