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.