Question

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

• A. (I) implies (IV)
• B. (I) implies (III)
• C. (II) implies (III)
• D. (II) implies (IV)

Hint:

Remember: Uniqueness of linear extensions is guaranteed by spanning sets, and existence of extensions is guaranteed by bases (spanning + independence).

Answer 1

The Correct Option is B, D

Step 1: Understanding property (I).
If every function \( f : S \to W \) extends to a linear map \( V \to W \), this is only possible when \( S \) is a basis of \( V \). Because defining a linear map on a basis uniquely determines its extension on the whole vector space. Thus, \( S \) must be linearly independent (property (III)) and spanning (property (IV)). Therefore, (I) ⇒ (III) and (I) ⇒ (IV) both hold true logically.

Step 2: Understanding property (II).
Property (II) means: If two linear maps agree on \( S \), they agree on the entire \( V \). This is true if and only if \( S \) spans \( V \). Thus, (II) ⇒ (IV).

Step 3: Connection between (II) and (III).
(II) does not ensure linear independence, only that \( S \) spans \( V \). Hence, (II) ⇒ (III) is false.

Final Answer: \[ \boxed{\text{(B) and (D)}} \]