Vector space/Basis/Exchange lemma/Fact

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

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

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

Then also the family

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

is a basis of ${\displaystyle {}V}$.