Let V be the vector space over R of all real-valued sequences (a„)n20· A real-valued sequence is called a Fibonacci sequence if it satisfies the recursion relation an+1 = an + an-1, for all n > 1. Let W be the set of all Fibonacci sequences. (a) Show that W is an R-vector subspace of V. (b) What is the dimension of W over R? (c) Let @ := 1+v5 2 be the golden ratio, and let ß := . Show that the sequences (")n20 and 1-/5 (B")n20 form a basis for W. (d) Let v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...) be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the previous two. Express v as a linear combination of the vectors (@")n20 and (B")n20, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
Let V be the vector space over R of all real-valued sequences (a„)n20· A real-valued sequence is called a Fibonacci sequence if it satisfies the recursion relation an+1 = an + an-1, for all n > 1. Let W be the set of all Fibonacci sequences. (a) Show that W is an R-vector subspace of V. (b) What is the dimension of W over R? (c) Let @ := 1+v5 2 be the golden ratio, and let ß := . Show that the sequences (")n20 and 1-/5 (B")n20 form a basis for W. (d) Let v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...) be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the previous two. Express v as a linear combination of the vectors (@")n20 and (B")n20, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let V be the vector space over R of all real-valued sequences (an)n20· A real-valued sequence is called a
Fibonacci sequence if it satisfies the recursion relation
an+1 = an + an-1,
for all n > 1.
Let W be the set of all Fibonacci sequences.
(a) Show that W is an R-vector subspace of V.
(b) What is the dimension of W over R?
1+v5
be the golden ratio, and let ß := . Show that the sequences (@" ),20 and
1-/3
( c) Let ω :=
(B")n20 form a basis for W.
(d) Let
v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...)
be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the
previous two. Express v as a linear combination of the vectors (@")n2o and (B")n20, and use this to
obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
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