Linear mapping/Determination on basis/Fact

Determination theorem for linear mappings

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}v_{i}$, ${}i\in I$, denote a basis of ${}V$, and let ${}w_{i}$, ${}i\in I$, denote elements in ${}W$.

Then there exists a unique

linear mapping

$f\colon V\longrightarrow W$

with

$f(v_{i})=w_{i}{\text{ for all }}i\in I.$

Write a proof