Difference between revisions of "12-240/Classnotes for Tuesday October 09"

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(theorems)
(theorems)
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<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>\all\!\,</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
  
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==

Revision as of 16:57, 12 October 2012

In this lecture, the professor concentrate on basics and related theorems.

Definition of basic

β \subset \!\, 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 \sum \!\, ai.ui = 0 ai \in\!\, F, ui \in\!\, β

\sum \!\, ai.ui = 0 = \sum \!\, 0.ui

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

Lecture notes scanned by Oguzhancan