Bases of Finitely Generated Vector Space have Equal Cardinality
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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): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.10$
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