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). 
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