## Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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?

-

## 1 Answer

For a 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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 27, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9295854568481445, "perplexity_flag": "head"}