If $S, T$ are subsets of a vector space, is $[S ∩ T ] = [S] ∩ [T ]$?

Question

Can someone help me visualize what is going on here? What is the difference between $[S ∩ T ]$ and $[S] ∩ [T]$?

Answer 1

On one hand: $$S \cap T \subseteq S \subseteq [S] \implies [S \cap T]\subseteq [S],$$and similarly $[S\cap T]\subseteq[T]$, hence $[S \cap T]\subseteq [S]\cap [T]$. In words: the linear span of the intersection of two sets is contained in the intersection of the linear span of the two sets.

The other inclusion is, in general, false. Take for example in $\Bbb R^2$: $$S = \{(1,0),(0,1)\}, \quad T = \{(1,1)\},$$so that: $$[S \cap T] = \{0\} \supsetneq [S]\cap [T] = \Bbb R^2 \cap \Bbb R(1,1) = \Bbb R(1,1).$$

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Obs.: I use $\Bbb R(1,1)$ as the set $\{\lambda(1,1) \mid \lambda \in \Bbb R\}$, that is, the line passing through the origin with direction $(1,1)$.

Answer 2

Assuming $ [] $ meaning $ span $ the statement is false:
Let $ S $ be $ \{(1,0)\} $ and let $ T $ be $ \{(2,0)\} $
$ [S \cap T] = span(S \cap T) = \emptyset $
while $ [S] \cap [T] = span(S) \cap span(T) = span(S) = span(T) = span\{(1,0)\} = span\{(2,0)\} $