Vector space Basis

// Vector space Basis //

Full PDF : Vector space Basis (GATE 2017)

Definition: "A vector space V is a collection of objects with a (vector)

addition and scalar multiplication defined that closed under both operations

and which in addition satisfies the following axioms" :

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

(ii) α(βx) = (αβ)x

(iii) x + y = y + x for all x, y ∈ V

(iv) x + (y + z) = (x + y) + z for all x, y, z ∈ V

(v) α(x + y) = αx + αy

(vi) ∃O ∈ V 􀁺 0 + x = x; 0 is usually called the origin

(vii) 0x = 0

(viii) ex = x where e is the multiplicative unit in F.