lecture7_web

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

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L7-1 EEL 6550 Noise in Linear Systems Lecture 7 I NTRODUCTION TO A BSTRACT A LGEBRA DEFN A binary operation * on a set G is a rule that for each a G, b G assigns c = a * b , such that c G . DEFN A group consists of a set G and a binary operation * with the following properties: 1. Associativity ( a * b ) * c = a * ( b * c ) for a, b, c G 2. Existence of Identity There exists e G such that a * e = e * a = a , for all a G 3. Existence of Unique Inverses For each a G , there exists a unique element a - 1 G such that a * a - 1 = a - 1 * a = e DEFN A group is said to be commutative or abelian if it also satisfies: 4. Commutativity : For all a, b G , a * b = b * a If a group is commutative, then the group operation is often represented as “ + Examples of groups: The set of integers forms a commutative group under addition . The set of integers does not form a group under multiplication . Why? The set of rational numbers excluding zero forms a group under multiplication .

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