12-240/Classnotes for Tuesday October 09: Difference between revisions
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In this lecture, the professor |
In this lecture, the professor concentrated on bases and related theorems. |
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== Definition of basis == |
== Definition of basis == |
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β <math>\subset \!\,</math> V is a |
β <math>\subset \!\,</math> V is a basis if |
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1/ It generates ( |
1/ It generates (span) V, span β = V |
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2/ It is linearly independent |
2/ It is linearly independent |
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== |
== Theorems == |
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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. |
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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 |
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Assume <math>\sum \!\,</math> |
Assume <math>\sum \!\,</math> ai∙ui = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β |
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<math>\sum \!\,</math> |
<math>\sum \!\,</math> ai∙ui = 0 = <math>\sum \!\,</math> 0∙ui |
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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 |
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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> |
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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>. |
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== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] == |
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] == |
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Latest revision as of 05:06, 7 December 2012
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In this lecture, the professor concentrated on bases and related theorems.
Definition of basis
β V is a basis if
1/ It generates (span) V, span β = V
2/ It is linearly independent
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.
proof: ( in the case β is finite)
β = {u1, u2, ..., un}
(<=) need to show that β = span(V) and β is linearly independent.
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 ai∙ui = 0 ai F, ui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in\!\,} β
ai∙ui = 0 = 0∙ui
since 0 can be written as a linear combination of elements of β in a unique way, ai=0 i
Hence β is linearly independent
(=>) every element of V can be written as a linear combination of elements of β in a unique way.
So, suppose ai∙ui = v = bi∙ui
Thus ai∙ui - bi∙ui = 0
(ai-bi)∙ui = 0
β is linear independent hence (ai - bi)= 0 i
i.e ai = bi, hence the combination is unique.
Clarification on lecture notes
On page 3, we find that then we say . The reason is, the Theorem 1.5 in the textbook.
Theorem 1.5: The span of any subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} of a vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . Moreover, any subspace of that contains must also contain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle span(S)}
Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} is a subset of , is a subspace of from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \subseteq span(\beta)} . From the "Moreover" part of Theorem 1.5, since is a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} containing , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle span(\beta)} must also contain .
Lecture notes scanned by Oguzhancan
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Lecture notes uploaded by gracez
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