Vector Space and Linear Transform ♣ Vector space, Subspace, Examples ♣ Null space, Column space, Row space of a matrix ♣ Spanning sets and Linear Independence ♣ Basis and Dimension ♣ Rank of a matrix ♣ Vector norms and matrix norms ♣ Linear transform • Pr

18 Definition of A Vector Space Definition: A vector space V (over R) is a set on which the operations of addition ⊕ and scalar multiplication are defined. The set V associated with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied. (A1) x ⊕ y = y ⊕ x ∀ x,y ∈ V (A2) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) ∀ x,y, z ∈ V (A3) ∃ 0 ∈ V such that x ⊕ 0 = x ∀ x ∈ V (A4) ∀ x ∈ V, ∃ − x ∈ V such that x ⊕ (−x) = 0 (A5) α (x ⊕ y) = (α x) ⊕ (α y), ∀ α ∈ R and x,y ∈ V (A6) (α + β) x = (α x) ⊕ (β x), ∀ α, β ∈ R and x ∈ V (A7) (α · β) x = α (β x), ∀ α, β ∈ R and x ∈ V (A8) 1 x = x for a 1 ∈ R and ∀ x ∈