Difference between revisions of "12-240/Classnotes for Thursday October 11"

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In this lecture, the porfessor concentrate on collaries of basics
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In this lecture, the professor concentrate on corollaries of basic and dimension.
 
== Annoucements ==
 
== Annoucements ==
 
TA Office Hours (Still pending!) @ 215 Huron St., 10th floor.
 
TA Office Hours (Still pending!) @ 215 Huron St., 10th floor.
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Topic: Replacement Theorem
 
Topic: Replacement Theorem
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== corollaries ==
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1/ If V has a finite basic β1, then any other basic β2 of V is also finite and |β1|=|β2|
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2/ "dim V" makes sense
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dim V =  |β| if V has a finite basic β
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Otherwise, dim V = <math>\infty \!\,</math>
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ex: dim P(F)= <math>\infty \!\,</math>
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3/ Assume dim V = n < <math>\infty \!\,</math> then,
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a) If G generate V then |G|<math>\ge \!\,</math> n & some set of G is a basic of V. ( If |G|= n, itself is a basic)
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b) If L is linearly independent then |L|<math>\le \!\,</math> n, if |L|=n then L is a basic, if |L|< n then L can be extended to become a basic.
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== Proofs ==
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1) β2 generate and β1 is linearly independent
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From replacement theorem
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|β2|<math>\ge\!\,</math> |β1| , ( role reversal), |β1|<math>\ge\!\,</math> |β2|
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Then |β2|= |β1|
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3) a) (|G| <math>\ge\!\,</math>  n)
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by dim V = n, exist basic β of V with n elements, Take L = β in the replacement lemma, |G| = n1
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|L| <math>\le\!\,</math> n1= |G| 
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Hence n <math>\le\!\,</math> |G|
  
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
 
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==

Latest revision as of 17:38, 12 October 2012

In this lecture, the professor concentrate on corollaries of basic and dimension.

Contents

Annoucements

TA Office Hours (Still pending!) @ 215 Huron St., 10th floor.

Peter - 11am - 1pm

Brandon 1pm - 3pm

Topic: Replacement Theorem

corollaries

1/ If V has a finite basic β1, then any other basic β2 of V is also finite and |β1|=|β2|

2/ "dim V" makes sense

dim V = |β| if V has a finite basic β

Otherwise, dim V = \infty \!\,

ex: dim P(F)= \infty \!\,

3/ Assume dim V = n < \infty \!\, then,


a) If G generate V then |G|\ge \!\, n & some set of G is a basic of V. ( If |G|= n, itself is a basic)

b) If L is linearly independent then |L|\le \!\, n, if |L|=n then L is a basic, if |L|< n then L can be extended to become a basic.

Proofs

1) β2 generate and β1 is linearly independent

From replacement theorem

|β2|\ge\!\, |β1| , ( role reversal), |β1|\ge\!\, |β2|

Then |β2|= |β1|

3) a) (|G| \ge\!\, n)

by dim V = n, exist basic β of V with n elements, Take L = β in the replacement lemma, |G| = n1

|L| \le\!\, n1= |G|

Hence n \le\!\, |G|

Lecture notes scanned by Oguzhancan