Question

For any subset \( U \) of \( \mathbb{R}^n \), let \( L(U) \) denote the span of \( U \). For any two subsets \( T \) and \( S \) of \( \mathbb{R}^n \), which one of the following statements is NOT true?

• A. If \( T \) is a proper subset of \( S \), then \( L(T) \) is a proper subset of \( L(S) \)
• B. \( L(L(S)) = L(S) \)
• C. \( L(T \cup S) = \{ u + v : u \in L(T), v \in L(S) \} \)
• D. If \( \alpha, \beta \) and \( \gamma \) are three vectors in \( \mathbb{R}^n \) such that \( \alpha + 2\beta + 3\gamma = 0 \), then \( L(\{ \alpha, \beta \}) = L(\{ \beta, \gamma \}) \)

Hint:

- The span of a set of vectors is the set of all possible linear combinations of those vectors.
- \( L(T \cup S) \) is not always a proper subset of \( L(S) \) if \( T \subseteq S \).

Answer 1

The Correct Option is A

1) Analyzing each option:
- (A) If \( T \) is a proper subset of \( S \), then \( L(T) \) is a proper subset of \( L(S) \): This is NOT true. The span of \( T \), \( L(T) \), could equal \( L(S) \) if the vectors in \( T \) span the same space as those in \( S \). Therefore, this is not always true.
- (B) \( L(L(S)) = L(S) \): This is true. The span of the span of \( S \) is just the span of \( S \).
- (C) \( L(T \cup S) = \{ u + v : u \in L(T), v \in L(S) \} \): This is true. The span of the union of two sets is the set of all possible linear combinations of elements from both sets.
- (D) If \( \alpha, \beta \) and \( \gamma \) are three vectors in \( \mathbb{R}^n \) such that \( \alpha + 2\beta + 3\gamma = 0 \), then \( L(\{ \alpha, \beta \}) = L(\{ \beta, \gamma \}) \): This is true. The linear span of \( \{ \alpha, \beta \} \) and \( \{ \beta, \gamma \} \) is the same because \( \alpha \) can be written as a linear combination of \( \beta \) and \( \gamma \).
Thus, the correct answer is (A).