Vector Space

   Let F be an orbitrary field then a non-empty set V is called a vector    space over the field F written as V(F) if following axioms are satisfied-
 

❶-There is defined an internal composition in V to be denoted additively such that (V,+) is an abelian group.

i.e.

1-Closure axiom:-

If α,β єV  then

α+βєV, ∀ α,βєBV

2-Associative law:-

If α,β,γєV then

α+(β+γ)=(α+β)+γ,∀α,β,γєV

3-Identity Element:

-

There Exist 0єV such that

α+0=0+α, ∀αєV

4-Inverse  axiom:-

Corresponding to each αєV there exist –αєV  such that

α+(-α)=(-α)+α=0, ∀αєV

5-commutative Law:-

If α+β=β+α, ∀ αєV

❷-There is defined an internal composition in V over F,called

scalar multiplication,i.e.

aєF,αєV͢→aαєV, ∀ aєF,αєV

❸-The two composition that is addition on V and scalar multiplication on V(F) satisfies following postulates-

१-a(α+β)=aα+bβ, ∀ α,βєV,aєF

२-(a+b)α=aα+bα,  ∀ αєV, a,bєF

३-(ab)α=a(bα),   ∀ a,bєF, αєV

 ४-1.α=α,  ∀ αєV and I is the unity element of F

Note-

❶-If F is a field of real number then V is called real vector space and is written as V(R).If F is a field of rational number or complex number then v is called rational vector space or Complex vector space.

❷-The  element of V are called vectors and elements of F as scalars.Here the word vector is not a physical quantity which have magnitude and direction,these are only elements of V.

❸-Vector space is also called as Linear space.

 
❹-In a vector space we deals with four composition two internal composition in the field F,one internal composition in V and one external composition in V(F).