12240/Classnotes for Thursday October 11

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