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

Latest revision as of 16:38, 12 October 2012

#	Week of...	Notes and Links
Additions to this web site no longer count towards good deed points.
1	Sep 10	About This Class, Tuesday, Thursday
2	Sep 17	HW1, Tuesday, Thursday, HW1 Solutions
3	Sep 24	HW2, Tuesday, Class Photo, Thursday
4	Oct 1	HW3, Tuesday, Thursday
5	Oct 8	HW4, Tuesday, Thursday
6	Oct 15	Tuesday, Thursday
7	Oct 22	HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
8	Oct 29	Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
9	Nov 5	Tuesday, Thursday
10	Nov 12	Monday-Tuesday is UofT November break, HW7, Thursday
11	Nov 19	HW8, Tuesday,Thursday
12	Nov 26	HW9, Tuesday , Thursday
13	Dec 3	Tuesday UofT Fall Semester ends Wednesday
F	Dec 10	The Final Exam (time, place, style, office hours times)
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In this lecture, the professor concentrate on corollaries of basic and dimension.

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

Peter - 11am - 1pm

Brandon 1pm - 3pm

Topic: Replacement Theorem

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| $\geq \!\,$ 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| $\leq \!\,$ n, if |L|=n then L is a basic, if |L|< n then L can be extended to become a basic.

1) β2 generate and β1 is linearly independent

From replacement theorem

|β2| $\geq \!\,$ |β1| , ( role reversal), |β1| $\geq \!\,$ |β2|

Then |β2|= |β1|

3) a) (|G| $\geq \!\,$ n)

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

|L| $\leq \!\,$ n1= |G|

Hence n $\leq \!\,$ |G|

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 == corollaries ==
+/ If V has a finite basic β1, then any other basic β2 of V is also finite and |β1|=|β2|
+/ "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>
+/ 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 ==
+) 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|
+) 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]] ==