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 = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty \!\,}

ex: dim P(F)= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty \!\,}

3/ Assume dim V = n < Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty \!\,} then,

a) If G generate V then |G|Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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|Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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|Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ge\!\,} |β1| , ( role reversal), |β1|$\geq \!\,$ |β2|

Then |β2|= |β1|

3)

Lecture notes scanned by Oguzhancan

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