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Vector Space (Theorem-5)

Labels: Linear Algebra, Vector Space Theorem

Statement:- The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are

( 1) α∊W, β∊W ⇨ α-β∊W

(2) a∊F, α∊W ⇨ aα∊W

Proof:- Necessary Condition:-

Let V be a vector space over the field F and W be its subspace.

∴ W will be a vector space over the same field F.

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Field & Integral Domain

Labels: Abstract Algebra, Field

Integral domain:-

1- A commutative ring with unity element and without zero divisors is called an integral domain.

2- An algebraic system (D,+,.) where D is a non-empty set with two binary compositions to be denoted by addition and multiplication is called an integral domain if following axioms are satisfied-

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Definition of Ring

Labels: Abstract Algebra, Ring

Ring:-

An algebraic system (R,+,.) consisting of a non-empty set R with two binary composition (to be denoted by addition and multiplication) is called a ring if following axiom are satisfied-

(R,+) is an abelian group.

1- Closure axiom

2- Associative law

3- Identity element

4- Inverse axiom

(R,.) is a semi group.

1- Closure axiom

2- Associative law

3- Multiplication distributive over addition i.e.

a.(b+c)=a.b+a.c, ∀ a, b, c∊R

and (b+c).a=b.a+c.a, ∀a, b, c∊R

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