12-240/Classnotes for Thursday October 11
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 =
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.
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|