MATH_2250_001_FALL_2023_PSET6
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University of Utah *
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Course
2250
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
7
Uploaded by tinalove801
University of Utah
Fall 2023
MATH 2250-001 PSet 6 Specification
Instructor: Alp Uzman
Subject to Change; Last Updated: 2023-10-17
1
Background
In this problem set, we will explore fundamental concepts in linear algebra: linear
independence versus linear dependence, spans, bases, and linear subspaces.
These
concepts not only form the backbone of linear algebra but are also pivotal in under-
standing and describing relations between functions; in particular solutions to linear
ODEs.
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.
A note on notation: commonly, vectors
v
are denoted either in bold
v
, with an
arrow
⃗v
, or both
⃗
v
. We’ve opted for minimal notation in this problem set, relying on
context to distinguish vectors from scalars. But you are welcome to use any notation
you’re comfortable with!
2.1
Linear Independence, Spans, Bases, Linear Subspaces
1. For each of the following two 2D vectors
u
= (
u
1
, u
2
)
,
v
= (
v
1
, v
2
)
, determine
whether they are linearly dependent or linearly independent.
Record also if
they constitute a basis of the vector space
R
2
.
(a)
u
= (0
,
2)
,
v
= (0
,
3)
(b)
u
= (0
,
2)
,
v
= (3
,
0)
(c)
u
= (2
,
2)
,
v
= (2
,
−
2)
(d)
u
= (2
,
−
2)
,
v
= (
−
2
,
2)
2. For each of the following three 2D vectors
u
= (
u
1
, u
2
)
,
v
= (
v
1
, v
2
)
,
w
=
(
w
1
, w
2
)
, express
w
as a linear combination of
u
and
v
.
Alp Uzman
Page 1 of 7
uzman@math.utah.edu
University of Utah
Fall 2023
(a)
u
= (5
,
7)
,
v
= (2
,
3)
,
w
= (1
,
1)
(b)
u
= (5
,
−
2)
,
v
= (
−
6
,
4)
,
w
= (5
,
6)
3. For each of the following three 3D vectors
u
= (
u
1
, u
2
, u
3
)
,
v
= (
v
1
, v
2
, v
3
)
,
w
= (
w
1
, w
2
, w
3
)
, determine whether they are linearly dependent or linearly
independent. If they are linearly dependent, find scalars
a
,
b
, and
c
not all zero
such that
au
+
bv
+
cw
= 0
. Record also if they constitute a basis of the vector
space
R
3
.
(a)
u
= (1
,
1
,
−
2)
,
v
= (
−
2
,
−
1
,
6)
,
w
= (3
,
7
,
2)
(b)
u
= (1
,
1
,
0)
,
v
= (5
,
1
,
3)
,
w
= (0
,
1
,
2)
4. For each of the following collection of vectors, determine whether or not they
constitute a basis. In each case, the number of entries of each vector indicate
the dimension of the ambient linear space, thus for instance for the vectors
(1
,
0
,
0
,
0
,
0)
,
(0
,
1
,
0
,
0
,
0)
,
(0
,
0
,
0
,
0
,
1)
the ambient space would be the 5D vector space
R
5
.
(a)
v
1
= (4
,
7)
,
v
2
= (5
,
6)
(b)
v
1
= (3
,
−
1
,
2)
,
v
2
= (6
,
−
2
,
4)
,
v
3
= (5
,
3
,
−
1)
(c)
v
1
= (1
,
7
,
−
3)
,
v
2
= (2
,
1
,
4)
,
v
3
= (6
,
5
,
1)
,
v
4
= (0
,
7
,
13)
(d)
v
1
= (3
,
−
7
,
5
,
2)
,
v
2
= (1
,
−
1
,
3
,
4)
,
v
3
= (7
,
11
,
3
,
13)
(e)
v
1
= (2
,
0
,
0
,
0)
,
v
2
= (0
,
3
,
0
,
0)
,
v
3
= (0
,
0
,
7
,
6)
,
v
4
= (0
,
0
,
4
,
5)
5. For each of the following collections
W
of vectors, determine whether or not
W
is a linear subspace.
(a)
W
is the set of all 3D vectors
(
x
1
, x
2
, x
3
)
such that
x
1
+
x
2
+
x
3
= 1
.
(b)
W
is the set of all 4D vectors
(
x
1
, x
2
, x
3
, x
4
)
such that
x
1
+ 2
x
2
+ 3
x
3
+ 4
x
4
= 0
.
(c)
W
is the set of all 4D vectors
(
x
1
, x
2
, x
3
, x
4
)
such that
x
1
= 3
x
3
and simultaneously
x
2
= 4
x
4
.
(d)
W
is the set of all 4D vectors
(
x
1
, x
2
, x
3
, x
4
)
such that
x
1
x
2
=
x
3
x
4
.
Alp Uzman
Page 2 of 7
uzman@math.utah.edu
University of Utah
Fall 2023
6. For each of the following systems of linear equations, find two 4D solution
vectors
u
= (
u
1
, u
2
, u
3
, u
4
)
and
v
= (
v
1
, v
2
, v
3
, v
4
)
such that the solution space is
the set of all linear combinations of the form
su
+
tv
where
s
and
t
are arbitrary
numbers. In linear algebra language one would say in this case that the vectors
u
and
v
together span the space of solutions.
Further, determine whether or
not the vectors
u
and
v
are linearly independent.
(a)
x
1
+ 3
x
2
+ 8
x
3
−
x
4
= 0
,
x
1
−
3
x
2
−
10
x
3
+ 5
x
4
= 0
,
x
1
+ 4
x
2
+ 11
x
3
−
2
x
4
= 0
(b)
x
1
+ 3
x
2
+ 2
x
3
+ 5
x
4
−
x
5
= 0
,
2
x
1
+ 7
x
2
+ 4
x
3
+ 11
x
4
+ 2
x
5
= 0
,
2
x
1
+ 6
x
2
+ 5
x
3
+ 12
x
4
−
7
x
5
= 0
7. For each of the following systems of linear equations, find a basis for the solution
space.
(a)
x
1
+ 3
x
2
+ 4
x
3
+ 5
x
4
= 0
,
2
x
1
+ 6
x
2
+ 9
x
3
+ 5
x
4
= 0
(b)
x
1
−
3
x
2
−
9
x
3
−
5
x
4
= 0
,
2
x
1
+
x
2
−
4
x
3
+ 11
x
4
= 0
,
x
1
+ 3
x
2
+ 3
x
3
+ 13
x
4
= 0
2.2
Linear Spaces of Functions
8. Find the constants denoted by capital letters in each of the following
partial
fraction decompositions
.
(a)
x
−
5
(
x
−
2)(
x
−
3)
=
A
x
−
2
+
B
x
−
3
(b)
2
x
(
x
2
−
1)
=
A
x
+
B
x
−
1
+
C
x
+ 1
(c)
8
x
(
x
2
+ 4)
=
A
x
+
Bx
+
C
x
2
+ 4
Alp Uzman
Page 3 of 7
uzman@math.utah.edu
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(d)
2
x
(
x
+ 1)(
x
+ 2)(
x
+ 3)
=
A
x
+ 1
+
B
x
+ 2
+
C
x
+ 3
(e)
x
2
−
29
x
+ 5
(
x
−
4)
2
(
x
2
+ 3)
=
A
x
−
4
+
B
(
x
−
4)
2
+
Cx
+
D
x
2
+ 3
(f)
x
3
+ 10
x
2
+ 3
x
+ 36
(
x
−
1)(
x
2
+ 4)
2
=
A
x
−
1
+
Bx
+
C
x
2
+ 4
+
Dx
+
E
(
x
2
+ 4)
2
9. Consider the ODE
dy
dx
= 0
.
We know that the only functions
y
=
y
(
x
)
whose derivative is zero are the
constant functions, that is, the general solution to the ODE is
y
(
x
) =
y
0
,
where
y
0
=
y
(0)
is the initial condition. Thus any solution to the given ODE
is a constant multiple of the function
β
0
(
x
) = 1
that is constantly
1
. In linear
algebra language, we may thus say that the linear space of the solutions to the
given ODE is one dimensional, and that the function
β
0
(
x
) = 1
is a basis of it.
Your job in this problem is to generalize this statement to higher order ODEs
of the form
d
n
y
dx
n
= 0
.
(a) First consider the case of
n
= 2
, i.e.
y
′′
= 0
.
We obtained the general solution to the earlier example by integrating both
sides; to find the general solution to the case of
n
= 2
, one now would thus
need to integrate both sides two times. Based on this, first verify that the
general solution is
y
(
x
) =
y
1
x
+
y
0
,
with
y
(0) =
y
0
and
y
′
(0) =
y
1
. Next, find two functions
β
0
(
x
)
and
β
1
(
x
)
such that
β
′′
0
= 0
,
β
′′
1
= 0
,
Alp Uzman
Page 4 of 7
uzman@math.utah.edu
University of Utah
Fall 2023
and moreover, given any particular solution
y
(
x
) =
y
1
x
+
y
0
, there are
unique
numbers
c
0
and
c
1
such that no matter what
x
is, one has
y
(
x
) =
c
1
β
1
(
x
) +
c
0
β
0
(
x
)
,
thus verifying that the space of solutions to
y
′′
= 0
is a 2D vector space,
and simultaneously, that the two functions
β
0
(
x
)
and
β
1
(
x
)
constitute a
basis for this space of solutions.
(b) After you are done with the previous part, now you can tackle the general
case. For
n
an arbitrary positive integer,
i. Compute the general solution to the ODE
y
(
n
)
= 0
.
Make sure to relate the constants to the
x
= 0
value as in the above
two ODEs.
ii. Find a basis for the linear space of solutions of the ODE consisting
of functions
β
0
(
x
)
,
β
1
(
x
)
, ...,
β
n
−
1
(
x
)
and verify that the functions
you chose do indeed constitute a basis by verifying that any particular
solution
y
(
x
)
to the ODE can be written as a linear combination
y
(
x
) =
c
n
−
1
β
n
−
1
(
x
) +
· · ·
+
c
1
β
1
(
x
) +
c
0
β
0
(
x
)
for some
unique
numbers
c
0
, c
1
, ..., c
n
−
1
.
You may find it useful to
tackle the case
n
= 3
first to make things more concrete!
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:
Alp Uzman
Page 5 of 7
uzman@math.utah.edu
University of Utah
Fall 2023
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.
•
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/SHNh2JwAroKKh3Lw9
.
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/3044717
,
see the Gradescope
documentation
for instructions.
Alp Uzman
Page 6 of 7
uzman@math.utah.edu
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5
When to Submit
This
problem
set
is
due
on
October
16,
2023
at
11:59
PM.
This problem set is due on October 23, 2023 at 11:59 PM.
Alp Uzman
Page 7 of 7
uzman@math.utah.edu