The title more or less says it all.... Let $V$ be a vector space (over your favorite field; $V$ not necessarily finite dimensional), and let $S$ be a subset of $V$. A maximal linearly independent subset of $S$ is exactly that: a subset of $S$ that is linearly independent yet not properly contained in any other linearly independent subset of $S$. (Equivalently, it is a basis for the subspace of $V$ that is spanned by $S$.)


Let $T$ be the intersection of all maximal linearly independent subsets of $S$. This $T$ might be as large as $S$, when $S$ itself is linearly independent. Alternatively, $T$ might be empty: if $\{v\_1,v\_2,v\_3\}$ is a basis for $V$, then both examples $S = {}${$v\_1,2v\_1$} and $S = {}${$v\_1,v\_2,v\_3,v\_1+v\_2+v\_3$} have corresponding $T=\emptyset$. There are plenty of intermediate cases as well: in the same notation, if $S = {}${$v\_1,v\_2,v\_3,v\_2+v\_3$} then $T={}${$v\_1$}.


Does this object $T$ have a standard name?



Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space?
For a [matroid](http://en.wikipedia.org/wiki/Matroid) the elements that are contained in every basis are called coloops, dual to the notion of a loop, which is an element not contained in any basis. Since you are interested in linearly independent sets perhaps adopting the language of matroids is not such a bad idea.


