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There are two ways to prove that every $k$-vector space $V$ has a basis. First, choose a linearly independent set of $V$, use Zorn lemma to prove that it must be contained by a maximal linearly independent set, then prove that the maximal linearly independent set is a basis.

Second, we can choose a generating set $B$ of $V$, prove that there is a minimal generating set which is contained in $B$, and then prove that every minimal generating set is a basis. However, I don't know how to prove that every generating set of a vector space has a minimal generating set, could someone help me with that?

Thank you!

Rushabh Mehta
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cbyh
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    Does this answer your question? An infinite generating set of a finite dimensional vector space contains a basis. Look at @AsafKaragila's answer in particular. – Rushabh Mehta Jan 22 '21 at 18:29
  • Are you assuming that your vector space is finite dimensional? – Arturo Magidin Jan 22 '21 at 18:32
  • You have to be careful; a direct application of a reverse Zorn's lemma need not work. For example, see here. – Arturo Magidin Jan 22 '21 at 18:35
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    In other words: the collection of generating subsets reverse-ordered by inclusion need not satisfy Zorn's Lemma. So you cannot hope to prove that every generating set contains a minimal generating set by applying Zorn's Lemma to that collection. The standard method is instead to consider the linearly independent subsets of your generating set, and prove that a maximal element of that collection spans and is a minimal spanning set. But that just turns you into the initial method in disguise. – Arturo Magidin Jan 22 '21 at 18:39
  • I am sorry, but I read an answer in that link (the one using Zorn lemma), and seems that it does not use the assumption that the space is finite-dimensional, so I am a little bit confused why it can not work for an arbitrary vector space. – cbyh Jan 22 '21 at 18:55
  • @cbyh: It does work in general; in fact, the answer explicitly says you do not need to assume that the space is finite dimensional. – Arturo Magidin Jan 24 '21 at 02:10

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