What is Vector Space ?

source: adobe stock

 Let V be a non-empty set and addition is an internal binary operation denoted 

by '+' that is a map from V x V → V which assigns to each ordered pair (x, y) ∈ V × V 

to a unique element of V denoted by x+y and multiplication is an external

 binary operation over a field F denoted ' . ' by that is a map from Fx V→ V which

 assigns to each ordered pair (α, x) ∈ F × V to a unique element of V denoted 

by α.x= αx, then V is called a (Left) vector space over the field F .

if the following conditions hold:

(i) (V, +) is an abelian group. The identity of the group (V, +) is called zero vector 

denoted by 0. 

(ii) α(x+ y) = αx+ αy for all x, y ∈V and α ∈ F 

(iii) (α+ β)x = αx + βx for all α, β ∈ F and x ∈ V 

(iv) α(βx) = (αβ)x = β(αx) for all α, β∈ F and x ∈ V 

(v) 1 · x = x for all x ∈ V 

Here 1 is the multiplicative identity of F.