In any vector space V over a field F,
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DEFINITION: Suppose that F is a field. A vector space V over F is a nonempty set with
two operations, “addition” and “scalar multiplication” satisfying certain requirements.
Addition is a map V × V −→ V : (v1,v2) −→ v1 + v2.
Scalar multiplication is a map F × V −→ V : (f,v) −→ fv.
The requirements are:
(i) V is an abelian group under the addition operation +.
(ii) f(v1 + v2) = fv1 + fv2 for all f ∈ F and v1, v2 ∈ V .
(iii) (f1 + f2)v = f1v + f2v for all f1, f2 ∈ F and v ∈ V .
(iv) f1(f2v) = (f1f2)v for all f1, f2 ∈ F and v ∈ V .
(v) 1F v = v for all v ∈ V .
Step-by-step explanation:
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