# Bases of Finitely Generated Vector Space have Equal Cardinality

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## Contents

## Theorem

Let $K$ be a division ring.

Let $G$ be a finitely generated $K$-vector space.

Then any two bases of $G$ are finite and equivalent.

## Proof

Since a basis is, by definition, both linearly independent and a generator, the result follows directly from Size of Linearly Independent Subset is at Most Size of Finite Generator.

$\blacksquare$

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 27$: Theorem $27.10$

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