"Vectors" can be anything at all. For example, consider the set of polynomials with real-valued coefficients. Is it a vector space over the reals? Check the axioms. What about polynomials with integer coefficients? (No, they are not closed under scalar multiplication).
Now, what does it mean for a set of polynomials (remember, just a kind of function) to be linearly independent? It's exactly the same definition as for n-tuples. If there's a collection of scalars, not all zero, one for each polynomial, such that the sum of each scalar times its polynomial is zero (in this case, zero means the zero polynomial, again a function!), then they are linearly dependent.
The point of this exercise is to get you to realize how general the vector space axioms are. Tuples of real numbers are the first vector space everyone sees, but the axioms can be satisfied by many, many kinds of objects. Just figure out what your scalars are, what the zero object is, and check the axioms carefully.