AXIOMS OF LINEAR SPACE

A triple (V, ⊕, ∗) is a linear space or a vector space over K, if the following four conditions (0) - (iii) hold: (0) V is a non-empty set and K is a field (K, +, ·) in abstract algebra sense (in most cases K = R or K = C). (i) It exists an internal binary operation in the set V (corresponding to "addition"), i.e. a function ⊕: V × V → V, which (therefore) associates to each ordered pair (u, v) ∈ V × V exactly one element u ⊕ v := ⊕(u, v) ∈ V. (ii) For the two sets K ja V exists a scaling function (corresponding to "multiplication by scalars"), i.e. a function ∗: K × V → V, (therefore) associates to each ordered pair (α, u) ∈ K × V exactly one element α∗u := ∗(α, u) &isin V. (iii) The internal binary operation and the scaling function have the following properties: A1. u ⊕ v = v ⊕ u for all u, v ∈ V (commutativity). A2. (u ⊕ v) ⊕ w = u ⊕ (v ⊕ w) for all u, v, w ∈ V (associativity). A3. There exists an element e ∈ V, for which u ⊕ e = u and e ⊕ u = u for all u ∈ V (neutral or zero element of addition). A4. For every u ∈ V there exists an element −u ∈ V, for which u ⊕ (−u) = e and (−u) ⊕ u = e (negative or inverse element). A5. α∗(u ⊕ v) = (α∗u) ⊕ (α∗v) for all α ∈ K, for all u, v ∈ V. A6. (α + β)∗u = α∗u ⊕ β∗u for all α, β ∈ K, for all u ∈ V. A7. α∗(β∗u) = (α·β)∗u kaikilla α, β ∈ K, for all u ∈ V. A8. 1∗u = u for all u ∈ V. If (V, ⊕, ∗) is a linear space over K, the set K is called the scalar space or scalar field. If K = R, the triple (V, ⊕, ∗) is a real linear space; if K = C, (V, ⊕, ∗) is a complex linear space. We usually talk briefly about linear space V.