Let V be a non-zero vector space over a field F. Let S ⊂ V be a non-empty set. Consider the following properties of S:
(I) For any vector space W over F, any map f : S → W extends to a linear map from V to W.
(II) For any vector space W over F and any two linear maps f, g : V → W satisfying f(s) = g(s) for all s ∈ S, we have f(v) = g(v) for all v ∈ V .
(III) S is linearly independent.
(IV) The span of S is V.
Which of the following statement(s) is /are true?
1
(I) implies (IV)
2
(I) implies (III)
3
(II) implies (III)
4
(II) implies (IV)