My recent interest in using transfinite induction in linear algebra has led me to pose a new question. (I will use c for the subset symbol)(adsbygoogle = window.adsbygoogle || []).push({});

Question: Use transfinite induction (not Zorn's lemma) to prove that if I is a linearly independent set and G is a set of generators (a spanning set) of an infinite-dimensional vector space V, with I c G, then there exists a basis B for V such that I c B c G.

There are not enough transfinite induction exercises in my set theory and topology textbooks so I had to pose this transfinite induction question myself. I'll give it a go:

Well-order G. Let A be the set of all vectors in G such that there exists a linearly independent set K where I c K c G and A c span(K) . A is non-empty since I c A, and thus there exists such a K. Suppose that the section S_v is a subset of A for some v in G. If v belongs to span(K), then by definition v belongs to A. If instead v does not belong to span(K), then KU{v} is a linearly independent set in V. Now I c KU{v} c G since I c K c G and v is in G. Furthermore,

AU{v} c span(KU{v}) since A c span(K). Thus v is in A. Thus A is an inductive subset of G. By the principle of transfinite induction, we have A = G. Consequently, there exists a linearly independent set B where I c B c G and G c span(B). So then V = span(G) c span(span(B)) = span(B), so that B is a basis for V.

How does this look?

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# Another infinite-dimensional basis question using transfinite induction

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