Linear independence/Test with linear forms/Fact

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Let ${\displaystyle {}v_{1},\ldots ,v_{n}\in V}$ be vectors. Suppose that for every ${\displaystyle {}k}$, there exists a linear form

${\displaystyle \varphi _{k}\colon V\longrightarrow K}$

such that

${\displaystyle \varphi _{k}(v_{k})\neq 0{\text{ and }}\varphi _{k}(v_{i})=0{\text{ for }}i\neq k.}$

Then the ${\displaystyle {}v_{1},\ldots ,v_{n}}$ are linearly independent.