MAT 223 Fall 2023
First Name Last Name Student ID Number email address (@mail.utoronto.ca)
Group Report 3
• Group reports are an opportunity for students to demonstrate what they learned. Students can
best demonstrate their learning by writing clear, thoughtful explanations. Don’t just give us
answers, show us you understand via your writing!
• Groups are assigned by TAs in tutorial subsections.
• Group size is 2 or 3 students. Exceptions can be made for special circumstances. Please contact
your TA.
• Work on each item individually first. Then meet with your group to create a final draft together.
• Learn how to submit work on gradescope and tag your group members. Go to
https://help.gradescope.com/article/m5qz2xsnjy-student-add-group-members
• One member submits for the whole group and tags their group mates.
• Use only your utoronto email, and search for your group members via their utoronto email.
• Groups can resubmit for full marks one time.
• Your tutorial TA is your point of contact regarding group reports. Please attend tutorials to get
help on group reports, get feedback, etc.
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MAT 223 Fall 2023
Productive Failure, Academic Integrity
Productive failure is an important aspect of MAT223 group reports. We allow one resubmission for full
marks. We encourage students to make honest efforts on groups reports so that you can get useful
feedback from your TA about your mathematical reasoning and writing, which will help you improve
and do better in future courses. We cannot help you learn if you copy work from others.
Academic honesty and integrity are fundamental to the mission of higher education and of the
University of Toronto.
It is okay to use appropriate outside resources. Please cite all your sources, including people, websites,
computational programs.
Your TA is your main point of contact for group reports. You will have time in tutorials to work on
group reports, students can get feedback on mathematics and writing before reports are turned in.
Academic dishonesty is...
• Using someone else’s words or ideas without proper documentation.
• Copying some portion of your text from another source without proper acknowledgement.
• Borrowing another person’s specific ideas without citation.
• Turning in an assignment written by another person, from an “service,” or copied from a
website.
• Using AI, such as ChatGPT, and presenting AI generated text as your own work without citation.
List the resources you used:
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MAT 223 Fall 2023
1. Let M =
3 3 −3 34 3 −4 1
2 1 −2 −1
.
(a) Find rref(M) explaining all of the steps.
(b) Describe the solutions to the system Mx⃗ = 0⃗ in terms of a span.
(c) Interpret the solutions to the system above in terms of the columns of M .
(d) Interpret the solutions to the system above in terms of the rows of M .
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MAT 223 Fall 2023
2. Suppose L :R3 →R3 is a linear transformation given defined by
L(v⃗)=
10 −19 106 −13 8
5 −13 9
 v⃗ ,
and letB =

11
1
 ,
12
3
 ,
42
1
 .
(a) Show thatB is a basis of R3.
(b) Write the matrix associated to L in the basisB.
(c) Interpret the linear transformation L geometrically.
(d) Discuss how the standard basis and the basisB are useful to describe the linear
transformation L.
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MAT 223 Fall 2023
3. Let T :R4 →R4 be a linear transformation. Suppose that a hyperplane in H ⊆R4 is described in
vector form by
x⃗ = r d⃗1+ sd⃗2+ t d⃗3+ p⃗.
The set T (H) is a translated span (also called affine space) in R4. List the possible cases of T (H)
and what conditions must be satisfied for each case.
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MAT 223 Fall 2023
4. Let L :R2 →R2 be the function defined by
L
([
x
y
])
=
[
x−p3yp
3x+ y
]
.
(a) Show that L is a linear transformation using the definition.
(b) Show that L is a linear transformation in another way (without using the definition).
(c) Describe the linear transformation using geometric terms. You can may use graphs to help
you convey your ideas. Write the explanation in a way that would make sense to someone
new to Linear Algebra.
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MAT 223 Fall 2023
5. For each item below, either construct a linear transformation satisfying the given condition or
explain why it is not possible:
(a) T : R2 →R2 such that ker(T )= Im(T ).
(b) T : R3 →R3 such that ker(T )= Im(T ).
(c) T : R4 →R4 such that ker(T )= Im(T ).
(d) Explain at least two conditions that a linear transformation T : Rn→Rm must satisfy so
that ker(T )= Im(T ).
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