Bases of Finitely Generated Vector Space have Equal Cardinality

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

• Dimension Theorem for Vector Spaces

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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