I've tried many things but have been unable to answer this question:
and are vectors in . Let be a linear combination of and have a single solution.
Prove are independent.
Hint: let be dependent vectors.
Thanks!
I've tried many things but have been unable to answer this question:
and are vectors in . Let be a linear combination of and have a single solution.
Prove are independent.
Hint: let be dependent vectors.
Thanks!
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 ?
Thanks nimon,
I don't study in English so I'm trying to translate as best as I can.
I still don't know where to continue with this:
and
I don't know which u vector belongs to. How do I know to replace it with or with ?
The vector we picked was any vector whose coefficient in the solution of (2) is non-zero, and we know that such a exists due to linear dependence. This could be any number between and , and we don't want to assume that it is or , we just know that one of them has non-zero coefficient so we let this be .
Given that , we can now replace in this expression with to get:
(3)
The notation means to sum over all but not .
Now just try and collect the coefficients in (3) to give as a linear multiple of
I hope this is helpful, and sorry for the lateness of my reply. Your English seems very good for someone who doesn't study it!