I think you haven't been given a reply until now because some of the parts of your question are hard to follow. You must ask your question so carefully that the reader doesn't have to guess what you mean: what do you mean by "a single solution"? and the hint surely should be: "Suppose that the above holds, but that are linearly dependent", not merely "Let be independent", because if we define these vectors to be independent, we cannot contradict ourselves! But if we suppose that they are dependent, then we can contradict the initial hypoethesis.
In this case, the proper statement of the problem should be: suppose that each has a unique representation as a linear multiple of the vectors Then prove that are linearly independent.
Proof (I will get you started)
Suppose that the statement holds and that are dependent, and let be written uniquely as
By linear dependence, there exist scalars such that
Then rearranging (2)
Now try to substitute this representation of
into the original representation of
given in (1)
. Is this a different expression for
as a linear combination of