Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$.
One fact is that the set of all functions $g\colon G\to K$ is a vector space over $K$ with the usual function operations. Hence one may speak of linear independence in such space.
However, jumping to Artin's theorem on the linear independence of characters, how can we speak about such a thing if characters are not functions as above and forms no obvious vector space?