Engineering Mathematics, Volume II, Second Edition

2

Linear Algebra

2.1 Concepts of Group, Ring, and Field

Definition 2.1. Let S be a non-empty set. Then a mapping f : S × S → S is called a binary operation in S.

A non-empty set along with one or more binary operations defined on it is called an algebraic structure.

Definition 2.2. A non-empty set G together with a binary operation f : G × G → G defined on it and denoted by * is called a group if the following axioms are satisfied:

(G₁) Associativity: For a, b, c, ∈ G,

 

(a * b) * c = a * (b * c)

 

(G₂) Existence of Identity: There exists an element e in G such that for all a ∈ G,

 

a * e = e * a = a

 

(G₃) Existence of Inverse Element: For each element a ∈ G, there exists an element b ∈ G, such that

 

a * b = b * a = e.

Definition 2.3. Let ...