Linear Vector Space

Abstract

For convenience, we begin by listing the axioms of a linear vector space once again. Let V be a set containing elements 0, x, y,… and Δ a field with elements 0, a, b, ⋯ , λ, μ, ⋯ . We say that V is a linear vector space over the ground field Δ if the following axioms are satisfied:

• (i) x ∈ V, y ∈ V ⇒ x + y ∈ V

• (ii) x + (y + z) = (x + y) + z

• (iii) x + 0 = x

• (iv) x + (−x) = 0

• (v) λ ∈ Δ, x ∈ V ⇒ λx ∈ V

• (vi) (λμ)x = λx_μx

• (vii) (λ + μ)x = λ(μx)

• (viii) λ(x + y) = λx + λy

• (ix) 1x = x; 1 is the unit of Δ

Some obvious consequences may be noted.