## Linear independence of a set of functions

I have a problem which I just can't get through:

Let $K$ be a field and let $f_1, ..., f_n$ be a set of functions belonging to the space $F(X,K)$ (i.e. the linear space of functions from a nonempty set $X$ to the field $K$ with a function constantly equal to $0$ as the zero element). Show that the set $f_1, ..., f_n$ is linearly independent if and only if there exist such elements $x_1, ..., x_n$ in the set $X$ that the vector family $\alpha_1, ..., \alpha_n$, where $\alpha_i = (f_i(x_1), f_i(x_2), ..., f_i(x_n))$ in the space $K^n$ is linearly independent.

I will be grateful for every suggestion.