Steinitz Exchange Lemma

Theorem: Steinitz Exchange Lemma

Let $B=\{{\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n\}$ be a basis of a finitely generated vector space $(V,K,+,\cdot )$.

For every set $\{{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m\}\subset V$ of linearly independent vectors there are $n-m$ vectors in $B$ (without loss of generality - ${\mathbf{v}}_{m+1},\cdots ,{\mathbf{v}}_n$) such that

$$\{{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m,{\mathbf{v}}_{m+1},\cdots ,{\mathbf{v}}_n\}$$

is also a basis of $(V,K,+,\cdot )$.

PROOF

TODO