Difference between revisions of "12-240/Classnotes for Thursday October 11"
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− | In this lecture, the | + | 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 | ||
+ | |||
+ | == 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 = <math>\infty \!\,</math> | ||
+ | |||
+ | ex: dim P(F)= <math>\infty \!\,</math> | ||
+ | |||
+ | 3/ Assume dim V = n < <math>\infty \!\,</math> then, | ||
+ | |||
+ | |||
+ | 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) | ||
+ | |||
+ | 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. | ||
+ | == Proofs == | ||
+ | 1) β2 generate and β1 is linearly independent | ||
+ | |||
+ | From replacement theorem | ||
+ | |||
+ | |β2|<math>\ge\!\,</math> |β1| , ( role reversal), |β1|<math>\ge\!\,</math> |β2| | ||
+ | |||
+ | Then |β2|= |β1| | ||
+ | |||
+ | 3) a) (|G| <math>\ge\!\,</math> n) | ||
+ | |||
+ | by dim V = n, exist basic β of V with n elements, Take L = β in the replacement lemma, |G| = n1 | ||
+ | |||
+ | |L| <math>\le\!\,</math> n1= |G| | ||
+ | |||
+ | 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
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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 =
ex: dim P(F)=
3/ Assume dim V = n < then,
a) If G generate V then |G| 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| 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| |β1| , ( role reversal), |β1| |β2|
Then |β2|= |β1|
3) a) (|G| n)
by dim V = n, exist basic β of V with n elements, Take L = β in the replacement lemma, |G| = n1
|L| n1= |G|
Hence n |G|