The discussion revolves around the relationship between the span of the intersection of two subsets, S_1 and S_2, of a vector space, and the intersection of their spans. Participants are exploring the conditions under which the equality span(S_1 ∩ S_2) = span(S_1) ∩ span(S_2) holds true, particularly focusing on the implications of the subsets being vector spaces.
The discussion is ongoing, with participants expressing differing views on the conjecture. Some have provided examples to challenge the initial assumptions, while others are prompted to clarify their understanding of the term "span" and its implications in this context. There is a call for more rigorous definitions and conditions to be included in the discussion.
Participants are encouraged to define terms clearly and consider the implications of their conjectures. There is an acknowledgment that the definition of "span" and the case of the empty set may need to be addressed in the context of the problem.
batballbat said:Homework Statement
S_1 and S_2 are subsets of a vector space. When is this:span(S_1 \cap S_2) = span(S_1) \cap span(S_2) true? Prove it.Homework Equations
The Attempt at a Solution
conjecture: iff the two subsets are vector spaces.
It's easy to see that your guess is wrong. Let ##\{e_1,e_2\}## be the standard basis of ##\mathbb R^2##. Let ##S_1=\{e_1\}## and ##S_2=\{e_1,e_2\}##. We have $$\operatorname{span}S_1\cap\operatorname{span} S_2 =\operatorname{span}(S_1\cap S_2)$$ but neither ##S_1## nor ##S_2## is a subspace.batballbat said:Homework Statement
S_1 and S_2 are subsets of a vector space. When is this:span(S_1 \cap S_2) = span(S_1) \cap span(S_2) true? Prove it.
Homework Equations
The Attempt at a Solution
conjecture: iff the two subsets are vector spaces.