Question Details

(Solved) Math 110 Homework Assignment 18 due date: 3, 2017 1. The Fibonacci Numbers are the numbers defined by F0 = 0, F1 = 1, and Fn = Fn1 +Fn2 for n 2. (So...


Hi, can someone help me with my linear algebra homework? (see questions in attach file)

1. The Fibonacci Numbers are the numbers defined by F0
= 0, F1
= 1, and Fn =
Fn−1+Fn−2
for n ≥ 2. (So for instance F2 = F1 +F0
= 1+0 = 1, F3 = F2 +F1
= 1+1 = 2,
and F4 = F3 + F2
= 2 + 1 = 3, etc.)
(a) Suppose that we set ~w
n
= (Fn, F
n−1
) for n ≥ 1, and A =

1 1
1 0

.
Show that the recursion relation above means that ~w
n+1 = A~wn
.
(b) Use the eigenvectors of A to find a formula for A
k
~w
1
for any k ≥ 0.
(c) Use the answer from (b) to find a formula for the n-th Fibonacci number Fn
.
(d) The Lucas Numbers are the numbers defined by L0
= 2, L1
= 1 and Ln =
Ln−1 + Ln−2
for n ≥ 2. Find a formula for the n-th Lucas number.
2. Let A be the matrix
A =

19 −14
21 −16

.
Find a formula for the entries of A
k
(for k ≥ 1). As a check, compute A
2
and A
3
and
to see if they match your formulas.
Hint: The first column of any 2 × 2 matrix is "where (1, 0) gets sent", therefore if
~w
0
= (1, 0), then the first column of A
k
is A
k
~w
0
.
1 2
3 4
L
W
3. The diagram at right shows a simple Win/Lose game. In each
turn, we are at one of positions 1, 2, 3, or 4, or we have either won
(W ) or lost (L). If we are at position W or L we stay there since
the game is over. If we're at any of the positions 1 through 4, then
during that turn we leave (with equal probability) along one of the
lines coming out of that position. In other words, if we're at position
1, then there's a 1/4 chance that we'll end up at L the next turn, a
1/4 chance of ending up at 2, a 1/4 chance of ending up at 3, and a
1/4 chance of ending up at 4.
On the other hand, if we're at position 3, then there's a 1/2 chance of ending up at
position 1 and a 1/2 chance of ending up at position 4 on the next turn.
1
(a) Write down the 6 × 6 transition matrix that tells us how to get from one turn
of the game to another. When writing down the matrix, let's use the order of
positions 1, 2, 3, 4, W , and then L.
If you write down the correct matrix, the eigenvectors should be: ~v
1
= (1, −1, −1, 1, 0, 0),
~v
2
= (3, 2, 2, 3, −5, −5), ~v
3
= (0, −4, 2, 0, 1, 1), ~v
4
= (5, 0, 0, −5, 1, −1), ~v
5
= (0, 0, 0, 0, 1, 0),
and ~v
6
= (0, 0, 0, 0, 0, 1).
(b) For each of the squares 1, 2, 3, and 4, work out the probability of winning if you
start on that square.
(c) Which starting square has the best chance of winning?
The matrix













1 3 0 5 0 0
−1 2 −4 0 0 0
−1 2 2 0 0 0
1 3 0 −5 0 0
0 −5 1 1 1 0
0 −5 1 −1 0 1













−1
=













1
5
−
1
5
−
2
5
1
5
0 0
1
10
1
15
2
15
1
10
0 0
0 −
1
6
1
6
0 0 0
1
10
0 0 −
1
10
0 0
2
5
1
2
1
2
3
5
1 0
3
5
1
2
1
2
2
5
0 1













may be useful in answering the question.

 


Solution details:
STATUS
Answered
QUALITY
Approved
ANSWER RATING

This question was answered on: Sep 05, 2019

PRICE: $15

Solution~000200145091.zip (25.37 KB)

Buy this answer for only: $15

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free copy from our tutoring website www.aceyourhomework.com (Deadline assured. Flexible pricing. TurnItIn Report provided)

Pay using PayPal (No PayPal account Required) or your credit card . All your purchases are securely protected by .
SiteLock

About this Question

STATUS

Answered

QUALITY

Approved

DATE ANSWERED

Sep 05, 2019

EXPERT

Tutor

ANSWER RATING

GET INSTANT HELP/h4>

We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.

You can also use these solutions:

  • As a reference for in-depth understanding of the subject.
  • As a source of ideas / reasoning for your own research (if properly referenced)
  • For editing and paraphrasing (check your institution's definition of plagiarism and recommended paraphrase).
This we believe is a better way of understanding a problem and makes use of the efficiency of time of the student.

NEW ASSIGNMENT HELP?

Order New Solution. Quick Turnaround

Click on the button below in order to Order for a New, Original and High-Quality Essay Solutions. New orders are original solutions and precise to your writing instruction requirements. Place a New Order using the button below.

WE GUARANTEE, THAT YOUR PAPER WILL BE WRITTEN FROM SCRATCH AND WITHIN YOUR SET DEADLINE.

Order Now