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From Zorn's lemma, basis can be thought of as a maximal independent set as well as minimum cover (covering all the vectors). Is this observation correct ? Can this observation be related to the usual notion of graph theory where an independent set is just complement of vertex cover.

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Well, there is some reason for the same naming occuring in both areas. – Hagen von Eitzen Aug 13 '13 at 9:48
It may be that by "cover" you mean what is more usually known as "span" or "generating set", and then yes: a basis of a linear vector space over a field is a maximal linearly independent set = minimal generating set(=minimal spanning set). – DonAntonio Aug 13 '13 at 12:14
Independent sets in graphs are also known as stable sets and as cocliques. It might be best not to read too much into the name. – Chris Godsil Aug 13 '13 at 14:18

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