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

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

I will be grateful for every suggestion.