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3. consider infinite sequences a0 a1 a2 where aj ...

# Question: consider infinite sequences a0 a1 a2 where aj ...

###### Question details

Consider infinite sequences a0, a1, a2, ... where ajR; we will denote such sequences as {aj}. Define addition of two sequences {aj} and {bj} by the sequence a0+b0, a1+b1, ... and scalar multiplication by r ∈ R as ra0, ra1, ra2, .... It can be then shown that the set of all infinite sequences over the field R form a Vector Space (you do not have to show this).

(a) Let S be the subset of sequences that satisfy an = an−1 + an−2 for all n ≥ 2. Prove that S is also a Vector Space.

(b) Let {fn} be the Fibonacci sequence 1, 1, 2, 3, 5, 8, .... Note that {fn} ∈ S. Find another sequence {cn} ∈ S such that the two sequences {{fn}, {cn}} form a basis for S.