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

Let be a field and let be a set of functions belonging to the space (i.e. the linear space of functions from a nonempty set to the field with a function constantly equal to as the zero element). Show that the set is linearly independent if and only if there exist such elements in the set that the vector family , where in the space is linearly independent.

I will be grateful for every suggestion.