12-240/Classnotes for Thursday October 11
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In this lecture, the professor concentrate on corollaries of basic and dimension.
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 = [math]\displaystyle{ \infty \!\, }[/math]
ex: dim P(F)= [math]\displaystyle{ \infty \!\, }[/math]
3/ Assume dim V = n < [math]\displaystyle{ \infty \!\, }[/math] then,
a) If G generate V then |G|[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \ge\!\, }[/math] |β1| , ( role reversal), |β1|[math]\displaystyle{ \ge\!\, }[/math] |β2|
Then |β2|= |β1|
3)
Lecture notes scanned by Oguzhancan
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