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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Loading...

Similar Threads for Cardinality basis infinite |
---|

A Infinite matrices and the Trace function |

I Diagonalization and change of basis |

I Measures of Linear Independence? |

B Tensor Product, Basis Vectors and Tensor Components |

**Physics Forums | Science Articles, Homework Help, Discussion**