Let V be the vector space of Problem 29 and let the subset W consist of those elements {x_n) of V such that X_n = X_n-1 + X_n-2 for n ≥  2. Thus a typical element of W is the Fibonacci sequence

(a) Show that W is a subspace of V.
(b) Prove that W is 2-dimensional.

Problem 29

Let V be the set of all infinite sequences {x_n} = {x₁, x₂, x₃, ....} of real numbers. Let addition of elements of V and multiplication by scalars be defined as follows:

and

Transcribed Image Text:

{1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...}.