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.

Also see

 * Dimension Theorem for Vector Spaces