Vector space/Basis/Exchange theorem/Fact

Basis exchange theorem

Let ${}K$ denote a field and let ${}V$ denote a ${}K$-vector space, and let a basis ${}b_{1},\ldots ,b_{n}$ be given. Let

$u_{1},\ldots ,u_{k}$

denote a family of linearly independent vectors in ${}V$.

Then there exists a subset

${}J=\{i_{1},i_{2},\ldots ,i_{k}\}\subseteq \{1,\ldots ,n\}=I\,$

such that the family

$u_{1},\ldots ,u_{k},b_{i},i\in I\setminus J,$

is a basis of ${}V$.

In particular, ${}k\leq n$.

Proof, Write another proof