Vector space/Basis/Exchange lemma/Fact

Basis exchange lemma

Let ${}K$ denote a field, let ${}V$ denote a ${}K$-vector space, and let a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ be given. Let ${}w\in V$ be a vector with a representation

${}w=\sum _{i=1}^{n}s_{i}v_{i}\,,$

where ${}s_{k}\neq 0$ for some fixed ${}k$.

Then also the family

$v_{1},\ldots ,v_{k-1},w,v_{k+1},\ldots ,v_{n}$

is a basis of ${}V$.

Proof, Write another proof