12-240/Classnotes for Tuesday October 09: Difference between revisions
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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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Hence β is linearly independent |
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(=>) every element of V can be written as a linear combination of elements of β in a unique way. |
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== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] == |
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] == |
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Revision as of 15:59, 12 October 2012
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In this lecture, the professor concentrate on basics and related theorems.
Definition of basic
β V is a basic if
1/ It generates ( span) V, span β = V
2/ It is linearly independent
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.
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 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.
Lecture notes scanned by Oguzhancan
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