12-240/Classnotes for Thursday October 11

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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 = \infty \!\,

ex: dim P(F)= \infty \!\,

3/ Assume dim V = n < \infty \!\, then,


a) If G generate V then |G|\ge \!\, 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|\le \!\, 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|\ge\!\, |β1| , ( role reversal), |β1|\ge\!\, |β2|

Then |β2|= |β1|

3) a) (|G| \ge\!\, n)

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

|L| \le\!\, n1= |G|

Hence n \le\!\, |G|

Lecture notes scanned by Oguzhancan