lecture7_web - L7-1EEL 6550 Noise in Linear Systems Lecture...

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Unformatted text preview: L7-1EEL 6550 Noise in Linear Systems Lecture 7INTRODUCTION TO ABSTRACT ALGEBRADEFNAbinary operation*on a setGis arulethat for eachaG, bGassignsc=a*b, such thatcG.DEFNAgroupconsists of asetGand abinary operation*with the followingproperties:1.Associativity(a*b)*c=a*(b*c)fora, b, cG2.Existence of IdentityThere existseGsuch thata*e=e*a=a, forallaG3.Existence of Unique InversesFor eachaG, there exists a uniqueelementa-1Gsuch thata*a-1=a-1*a=eDEFNA group is said to becommutativeorabelianif it also satisfies:4.Commutativity: For alla, bG,a*b=b*aIf a group is commutative, then the group operation is often represented as +Examples of groups:Theset of integersforms a commutative group underaddition.Theset of integersdoesnotform a group undermultiplication. Why?Theset of rational numbers excluding zeroforms a group undermultiplication.Theset of (nn) matrices with real elementsforms a commutative group undermatrixaddition....
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This note was uploaded on 03/20/2012 for the course EEL 6650 taught by Professor Shea during the Spring '12 term at University of Florida.

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lecture7_web - L7-1EEL 6550 Noise in Linear Systems Lecture...

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