Linear independence/Test with linear forms/Fact

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

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

such that

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

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

Write a proof