Vector space/Finitely generated/Length of every basis/Fact/Proof

Proof

Let ${}{\mathfrak {b}}=b_{1},\ldots ,b_{n}$ and ${}{\mathfrak {u}}=u_{1},\ldots ,u_{k}$ denote two bases of ${}V$. According to the basis exchange theorem, applied to the basis ${}{\mathfrak {b}}$ and the linearly independent family ${}{\mathfrak {u}}$, we obtain ${}k\leq n$. When we apply the theorem with roles reversed, we get ${}n\leq k$, thus ${}n=k$.

To fact