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

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(Definition of basic)
(theorems)
Line 9: Line 9:
  
 
== 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.
  
 +
proof: ( in the case β is finite)
 +
 +
β = {u1, u2, ..., un}
 +
 +
(<=) need to show that β = span(V) and β  is linearly independent.
  
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==

Revision as of 16:38, 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.

Lecture notes scanned by Oguzhancan