Hi jayshizwiz,

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

**(1)**

By linear dependence, there exist scalars such that

**(2)**

with some

Then rearranging

**(2) **gives

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

?