I am reading "The linear algebra a beginning graduate student ought to know" by Golan, and I encountered a puzzling statement:(adsbygoogle = window.adsbygoogle || []).push({});

Let V be a vector space (not necessarily finitely generated) over a field F. Prove that there exists a bijective function between any two bases of V. Hint: Use transfinite induction.

If V is generated by a finite set (with n elements), then I know how to prove that any basis has at most n elements, and thus all bases will have the same number of elements. But for infinite-dimensional vector spaces, I'm confused. How do I use transfinite induction to prove that there is a bijective correspondence between two bases of V if V is infinite-dimensional?

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# Cardinality of a basis of an infinite-dimensional vector space

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