I have corrected a stupid mistake in (b). V is not 2-d, but 1-d, because the zero sequence (0) is also proportional to (F_i) , with the proportional factor being zero, so V is just a 1-d vector space consisting of all sequences that is proportional to the Fibonacci sequence, and we can just take the Fibonacci sequence as its basis.
For (c), I have to say that it is still problematic even with the two corrections: 1. we change F_1 into F_i and 2. we ignore the situation of (0, 0, 0… 0). T will not be a linear operator at all, since it will map every vector in V to the golden ratio (except for the zero sequence, whose image cannot be defined).
It is not hard to see. If we take the sequence to be the Fibonacci sequence, we have 1, 1, 2, 3, 5, 8, … the the ratio between two consecutive numbers approach the golden ratio, but if we double every number in the sequence to get 2, 2, 4, 6, 10, 16, … after division, the factor 2 in the denominator and the numerator will cancel, so the ratio between two consecutive numbers is exactly the same as before and so the limit will still be the golde ratio. It means that T(2(F_i)) is not twice T((F_i)) which violates linearity.
Ask your instructor about this and see his response.
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I have done the (b) part.
There are multiple problems with the (c) part. I strongly recommend you reach out to your instructor and ask him/ her for corrections.
