Bases of Finitely Generated Vector Space

Theorem

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

Then any two bases are finite and have the same number of vectors.

Proof

Since a basis is both linearly independent and a generator, this follows directly from Linearly Independent Subset of Finitely Generated Vector Space.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 27$: Theorem $27.10$
• For a video presentation of the contents of this page, visit the Khan Academy.