MATH_2250_001_FALL_2023_PSET9
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University of Utah *
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Course
2250
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by tinalove801
University of Utah
Fall 2023
MATH 2250-001 PSet 9 Specification
Instructor: Alp Uzman
Subject to Change; Last Updated: 2023-10-30
1
Background
This problem set focuses on the fundamentals of spectral theory, namely eigenvalues
and eigenvectors in linear algebra. You’ll also dive into diagonalization, a technique
that, when applicable, simplifies the analysis of linear transformations. These con-
cepts have far-reaching applications in virtually all STEM fields. Through a variety
of examples, you’ll develop a deeper understanding of these mathematical tools and
their relevance in solving real-world problems.
2
What to Do
Solve the problems below. Though they may seem long, the additional text is meant
to guide you. When documenting your solutions, be thorough. Your goal is not just
to find the answer, but to create a clear, logical pathway to it that you or anyone else
could follow in the future.
1. Find the eigenvalues and eigenvectors of each of the following
2
×
2
matrices:
(a)
3
1
1
3
(b)
2
1
1
1
(c)
1
2
0
1
(d)
1
3
√
2
3
√
2
2. Consider the family of
stochastic matrices
A
=
p
1
−
q
1
−
p
q
Alp Uzman
Page 1 of 4
uzman@math.utah.edu
University of Utah
Fall 2023
parameterized by two numbers
0
< p <
1
and
0
< q <
1
.
Compute all
eigenvalues and eigenvectors of
A
. It’s likely that your final answers will depend
on
p
and
q
!
3. Consider the following four matrices:
I
:
A
=
5
−
3
2
0
II
:
A
=
5
−
4
3
−
2
III
:
A
=
5
1
−
9
−
1
IV
:
A
=
11
9
−
16
−
13
For each of the above matrices, determine whether or not it is diagonalizable.
If it is, find a diagonalizing matrix
P
and a diagonal matrix
D
such that
P
−
1
AP
=
D.
4. Consider a predator-prey population consisting of the foxes and rabbits living
in a certain forest. Initially, there are
F
0
foxes and
R
0
rabbits; after
n
months,
there are
F
n
foxes and
R
n
rabbits. We assume that the transition from each
month to the next is described by the equations
F
n
+1
= 0
.
6
F
n
+ 0
.
5
R
n
,
R
n
+1
=
−
rF
n
+ 1
.
2
R
n
,
where the parameter
r >
0
is the “capture rate” representing the average number
of rabbits consumed monthly by each fox.
(a) If
r
= 0
.
16
, compute that in the long term the populations of foxes and
rabbits stabilize, with 5 foxes for each 4 rabbits.
(b) If
r
= 0
.
175
, compute that in the long term the populations of foxes and
rabbits both die out.
(c) If
r
= 0
.
135
, compute that in the long term the fox and rabbit populations
both increase at the rate of 5% per month, maintaining a constant ratio
of 10 foxes for each 9 rabbits.
Alp Uzman
Page 2 of 4
uzman@math.utah.edu
University of Utah
Fall 2023
3
ChatGPT Regulations
This section is in case you decide to use ChatGPT in this problem set. If you will
not be using ChatGPT you may skip this section.
3.1
ChatGPT Versions
•
You may use either the GPT3.5 (freely available with a ChatGPT account) or
GPT4 (available with a ChatGPT Plus account).
•
Turn off all Custom Instructions before you start a chat. If you don’t, it will
be apparent in the archived version that you didn’t.
3.2
Chat Guidelines
•
Your first message in any given chat must be the following guardrails paragraph;
you may copy and paste it:
Hello. I am working on a differential equations and linear algebra
problem as part of a university class. My instructor has permitted
the use of ChatGPT, but only under specific guidelines to
encourage independent critical thinking. Please assist me by asking
probing questions, encouraging reflection, and providing general
insights about the concepts involved. Do not offer direct hints,
strategies, solutions, or step-by-step guidance. I seek to understand
the underlying principles and want to develop my own approach to
the problem. Your role is to facilitate my learning process without
directly leading me to the answer. Thank you!
•
You may copy and paste parts of this specification document, as well as parts
of the textbook or other sources.
•
You may not ask ChatGPT to write for you the solution for any one of the
problems in complete detail.
3.3
Archiving Chats
•
Once you are done with a chat with ChatGPT, click the "Share chat" icon on
the top righthand corner; see the
documentation
for details. In your chat don’t
include any personal information, and keep your user name hidden when you
are creating a link for anonymity.
•
Next you will use
Wayback Machine
to take a "snapshot" of your chat, see the
documentation
for the "Save Page Now" feature. You do not need an
Internet
Archive
account to do this, but having such an account (which is free) would
provide you with further options.
Alp Uzman
Page 3 of 4
uzman@math.utah.edu
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•
You have to take a snapshot of each one of your relevant chats separately, and
share the links to their archived versions in the form for this problem set.
•
To see an example of the outcome, see the Acknowledgements section in the
course syllabus. Note that the staff did use Custom Instructions in this case.
4
How to Submit
•
Step 1 of 2:
Submit the form at the following URL:
https://forms.gle/KiFkL4ZkN26sRq4U8
.
Your submission on Gradescope will receive a zero if you skip this step.
•
Step 2 of 2:
Submit your work on Gradescope at the following URL:
https://www.gradescope.com/courses
/565427/assignments/3044720
,
see the Gradescope
documentation
for instructions.
5
When to Submit
This problem set is due on November 6, 2023 at 11:59 PM.
Alp Uzman
Page 4 of 4
uzman@math.utah.edu