Link between 'minimal' spanning set in Linear Algebra & 'minimal' in Set Theory Theorem: Let $ V $ be a vector space and $ X = (v_1,...,v_s) \subseteq V $ be a subsequence of vectors. Then $ Span(X) $ is the minimal subspace that contains $X$.
Proof (what is written in my notes ):

*

*[ we show $ Span(X) $ is indeed a subspace of $ V $ ]


*We will prove minimality. Let $ U \subseteq V $ be a linear subspace s.t. $ X \subseteq U $ . Then also any linear combination of the vectors in $ X $ is in $ U $ ( since $ U  $ is a subspace ), hence $ Span(X) \subseteq U $. $ \Box $
In "How to prove it" by Velleman, there's the following definition of a minimal element of a set:

And later he adds the following:

I wanted to know if there is any connection/link between the Theorem above in linear algebra and the Definition I've provided that relates to Set Theory ( or are they totally different beings? ). If so, can you please explain? Relating to the defintion above, Is $ Span(X)$ a R-minimal element of some relation R? and what is the set $ B $ here s.t. $ Span(X) \in B $?
 A: There is indeed a connection, although there's a slight terminological mis-match: "minimal" in the original context corresponds to Vellman's "smallest," not Vellman's "minimal."
Specifically, the linear algebra result says that $Span(X)$ is smallest with respect to the subset relation amongst the collection of subspaces of $V$ containing $X$. That is:

*

*$B$ is the set of all subspaces $S$ of $V$ such that $X\subseteq S$, and


*$R$ is just $\subseteq$.
Note that as far as the choice of $R$ goes, this is exactly an instance of the situation described by Vellman's subsequent remark ("When comparing ..."). And elaborating on Vellman's remark, a phrase of the form "the minimal/smallest/least [structure]" generally means "the $\subseteq$-smallest element of the collection of all [structure]s." The only discrepancy is around the use of "minimal" rather than "smallest" (or "least," to mention another synonym); annoyingly, mathematicians aren't always careful about this, and when a term like "minimal" is used you should make sure the precise meaning is clear from context.

$\color{red}{\mathsf{WARNING}}$: keep in mind that minimal/smallest objects need not always exist! For example, if $X$ is finite and $V$ is infinite-dimensional, the set of infinite-dimensional subspaces of $V$ containing $X$ will not have a $\subseteq$-minimal element, let alone a $\subseteq$-smallest element.


(As an aside, it's worth noting that this characterization of $Span(X)$ does not depend on $X$ being finite: for every $X\subseteq V$ we can define $Span(X)$ to be the set of finite linear combinations of elements of $X$, and then $Span(X)$ is the smallest subspace of $V$ containing $X$ in exactly the same sense.)
