# 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 ):

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

2. 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. $$Span(X) \in B$$?

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.)