This document provides information about the Advanced Algebra I course taught by Melisa Vivanco during the 2021-1 semester. The course will be held virtually and cover topics such as set theory, combinatorics, vector spaces, matrices, determinants, and linear equation systems. Students will complete homework assignments, four midterm exams, and a final exam. Virtual classes and meetings will be held weekly, and tutoring is available online. The course aims to introduce abstract algebraic concepts and provide theoretical tools to support learning in other mathematics courses.

1. Advanced Algebra I
Melisa Vivanco
Semester 2021-1
[Original version in Spanish: Here]
Class: Mon, Tue, Fri 16–17 Site: Classroom
Tutorship: Wed 16–17 E-mail: jeronimozc@ciencias.unam.mx
Tutorship: Thr 16–17 E-mail: jruben@ciencias.unam.mx
Website: Melisa Vivanco E-mail: melisa.viva@ciencias.unam.mx
Course description
The purpose of this course is to introduce students to the discipline of algebra from a more
abstract perspective, in which the objectives are mainly directed towards the study of differ-
ent algebraic structures, rather than the simple resolution of particular equations (such as it is
common in high school education systems).
Furthermore, the course aims to be an additional course to provide students with theoretical
tools that facilitate their performance in other subjects on the curriculum map. For example, the
principles of set theory will promote familiarity with the predominant language in which a large
part of the proofs that the students will be faced with throughout their career. The presentation
of vector spaces, matrices, determinants, and systems of general equations will facilitate under-
standing later topics, not only in linear algebra courses but also in analytical geometry courses.
To mention one last example, studying the algebraic structures of real numbers and complex
numbers (corresponding to Advanced Algebra II) will support a large part of the results pre-
sented in the calculus, analysis, and complex variable courses.
All course topics are presented in a formal but introductory way. This approach enables the
students to later delve into different branches of mathematics.
List of topics
1. Foundations
• Intuitive notion and identity of sets
• Subsets; Empty set; Universal set
• Operations with sets; union; intersection; complement and difference
• Power set; Cartesian product; Set families
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2. Advanced Algebra I
• Relations (domain, codomain and range)
• Equivalence relations and partitions
• Functions (ranges and inverse ranges)
• Composition of functions; Inverse function
• Injective, surjective and bijective functions (cardinality)
• Numerical and algebraic structures; Induction Principle
2. Combinatorics
• Partitions (intuitive version)
• Cartesian counting sets (intuitive version)
• Permutations (intuitive version)
• Combinations
• Partitions (formal version)
• Cartesian counting sets (formal version)
• Permutations (intuitive version)
• Combinations and binomial coefficients
3. Vector Spaces
• Vectors and their operations
• The vector space Rn
• Vector subspaces
• Linear combinations; Linear dependence and independence
• The subspace generated by an arbitrary set of vectors
• Bases; Dimension
4. Matrices and Determinants
• Matrices: Definition and operations; Transpose of a matrix
• Elementary operations; Staggered matrices; Rank of a matrix
• The determinant of a square matrix: definition and properties
• Methods to calculate determinants.
• Characterization of the rank of a matrix by the determinant
5. Linear Equations Systems
• Preliminaries
• Existence of solutions
• Systems of n equations and n variables
• Homogeneous systems linear equations
• Associated homogeneous systems
• Systems solution method
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3. Advanced Algebra I
Required Material
Álgebra Superior, Alejandro Bravo Mojica; Hugo Rincón Meja and César Rincón Orta, Las
prensas de Ciencias, 2012
Álgebra Superior I, Antonio Lascurain Orive, Las prensas de Ciencias, 2012
Álgebra Superior, Humberto Cárdenas, Emilio Lluis, Francisco Raggi and Francisco Tomás,
Trillas, 1973
Complementary Material
Sets, Logic and Numbers, Clayton W. Dodge, Boston: Weber & Schmidt, 1973
Linear Algebra, Stephen H. Friedberg; Arnold J. Insel & Lawrence E. Spencer, Prentice Hall,
4th ed., 2003
An Introduction to Algebraic Structures, Joseph Landin, Dover, 1989
Course goals
1. Develop skills of mathematical language through the use of classical logic and set theory
2. Analyze combinatorial processes from abstract concepts, such as combinations, partitions
and permutations
3. Master the notions of relation and function, emphasizing the different properties that define
them
4. Understand the structural relations between matrices, determinants and Rn
5. Apply the results learned to the solution of systems of general equations
Class dynamic
Video class: Before the classes corresponding to bf Monday and bf Wednesday of each
week, I will send you each day, via our Classroom, a link to a prerecorded video of the
related class (this will further avoid unforeseen incidents due to operational problems re-
lated to sites, systems, connection, etc.). It is the student’s obligation to have seen the video
before 5 pm on the day in question.
Video meetings: On Friday (working) of each week we will have a video meeting to follow
up on the contents of the video classes corresponding to the previous Monday and Wednes-
day. It is very important that you attend and bf participate in these meetings. The Meet
link is available on the main page of the Virtual Room. (Due to the circumstances and by
virtue of carrying out a constant dynamic in the evolution of learning, the attendance and
participation of each student will be considered for the evaluation.)
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4. Advanced Algebra I
Virtual tutorships: Students will have access to video tutorships with the teaching assistant
every Tuesday and Thursday during the time assigned to the course. During these meetings
they can consult their doubts about the homework exercises, the evaluations and, in general,
about the course contents.
Assignments: Homework will be composed of exercises similar to those that will contain
the partial exams. There will be one assignment for each section from 1 to 4. The submission
of at least three of these assignments is mandatory to pass the course. The document with
the exercises will be sent to you via the Virtual Room through the section Homework. In
this same section you will find a window with the option it Submit an assignment. In this
box you should attach the document with the solutions to the exercises.
1. Assignments should be submitted in .pdf or .docx formats.
2. In case you cannot type them in Word or Latex, you are allowed to write them down
by hand and upload a photograph meeting the following conditions:
– The text must be legible
– The photograph is of good quality
– The name of the student is written
– The question is written before each answer
Graded assignments will be returned by the same means. In each case, the dates will be
specified in the section Plank in Classroom.
Home–exam: This assignment will correspond to section 5 of the course. The same condi-
tions apply as for ordinary assignments, but it will also cover the partial exam correspond-
ing to section 5
Midterms: There will be four ordinary partial exams: each corresponding to a section from
1 to 4. The exam will be proctored in real time by videoconference on Meet. The document
with the questions will be published in the Virtual Room. The students will have one hour
to solve it and send the photograph in the Homework section. (Review exam policies in the
corresponding section below.)
If the student is not satisfied with the assessment from the midterms, they will have the
option of taking a final exam that will cover the contents of the partial exams 1–4. If they
are not satisfied with this grade, they will have the option of taking a second final exam,
corresponding to all the topics of the course. This exam will be proctored in a session of
Meet with the videocamera on and may be partially written and partially oral. To be entitled
to either of the two final exams, the student must have turned in all the assignments, as
well as have complied with 80% attendance at the virtual meetings.
Assessment
Four midterms, 10% each (40% total)
Four assignments, 10% each (40% total)
Home–exam 10%
Attendance and participation 10%
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5. Advanced Algebra I
Course policies
Attendance policy
A record of attendance will be kept for virtual meetings on Thursdays. Virtual assistance to consultancies is not
absolutely mandatory. However, it is recommended to be present to maintain the learning pace and to participate
to achieve a constructive group dynamic. Attendance and participation in all virtual meetings will be taken into
consideration for the final grade.
Policies for late submission of exams and assignments
Delayed assignments will be accepted without penalty only in the case in which the justifying cause is communicated
to the professor or the teaching assistant by the due date. After this date, the assignments will be accepted up to
one week late with a 50 % reduction in the grade obtained. After this date, the grade assigned to the corresponding
assignment will be 0.
For taking the exams, the student must be present at the corresponding videoconference session. These sessions
could be divided into two parts for each exam, depending on the size of the group. In case that due to force majeure,
the student cannot be online to take the exam, she must notify the teacher or assistant as soon as possible. In this
case, an individual virtual meeting will be arranged, with the videocamera on, and the exam may be partially written
and partially oral.
Academic integrity and honesty
Students must subscribe to the academic policies of the General Statute of the UNAM established in the document
Derecho Universitario de los Alumnos. This includes cases of identity theft and plagiarism.
Accommodations for Disabilities
Reasonable accommodations will be made for students with verifiable disabilities.
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6. Advanced Algebra I
Tentative schedule
The topics and calendar may be subject to modifications by dispositions of the university, the faculty (for example,
regarding the dates for the final exams) or by circumstances related to the development of the course.
Week 01, 28/09 - 02/10: Foundations
• Mon 28: Presentation meeting (via Zoom)
• Wed 30: Video Class 1
• Fri 02: Video meeting 1
Week 02, 05/10 - 09/10: Foundations
• Mon 05: Video Class 2
• Wed 07: Video Class 3
• Thr 08: Video tutorship (via Meet)
• Fri 09: Video meeting 2 (via Meet)
Week 03, 12/10 - 16/10: Foundations
• Mon 12: Video Class 4
• Tue 13: Video Class 5
• Wed 14: Video tutorship (via Meet)
• Thr 15: Video tutorship (via Meet)
• Fri 16: Video meeting 3 (via Meet)
Week 04, 19/10 - 23/10: Combinatorics
• Mon 19: Video Class 6
• Tue 20: Video Class 7
• Wed 21: Video tutorship (via Meet)
• Thr 22: Video tutorship (via Meet)
• Fri 23: Video meeting 4 (via Meet); Submit Assignment 1 (via Homework/Classroom)
Week 05, 26/10 - 30/10: Combinatorics
• Mon 26: Video Class 8
• Tue 27: Video Class 9
• Wed 28: Midterm 1, proctored on Meet, upload to Homework
• Thr 29: Video tutorship; review Midterm 1
• Fri 30: Video meeting 5 (via Meet)
Week 06, 02/11 - 06/11: Combinatorics
• Mon 02: Holliday
• Tue 03: Video Class 10
• Wed 04: Video tutorship (via Meet)
• Thr 05: Video tutorship (via Meet)
• Fri 06: Video meeting 6 (via Meet) y Submit Assignment 2 (via Homework/Classroom)
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7. Advanced Algebra I
Week 07, 09/11 - 13/11: Vector Spaces
• Mon 09: Video Class 11
• Tue 10: Video Class 12
• Wed 11: Midterm 2, proctored on Meet, upload to Homework
• Thr 12: Video tutorship; review Midterm 2
• Fri 13: Video meeting 7 (via Meet)
Week 08, 16/11 - 20/11: Vector Spaces
• Mon 16: Holliday
• Tue 17: Video Class 13
• Wed 18: Video tutorship (via Meet)
• Thr 19: Video tutorship (via Meet)
• Fri 20: Video meeting 8 (via Meet)
Week 09, 23/11 - 27/11: Vector Spaces
• Mon 23: Video Class 14
• Tue 24: Video Class 15
• Wed 25: Video tutorship (via Meet)
• Thr 26: Video tutorship (via Meet)
• Fri 27: Video meeting 9 (via Meet) y Submit Assignment 3 (via Homework/Classroom)
Week 10, 30/11 - 04/12: Matrices and Determinants
• Mon 30: Video Class 16
• Tue 01: Video Class 17
• Wed 02: Midterm 3, proctored on Meet, upload to Homework
• Thr 03: Video tutorship; review Midterm 3
• Fri 04: Video meeting 10 (via Meet)
Week 11, 07/12 - 11/12: Matrices and Determinants
• Mon 07: Video Class 18
• Tue 08: Video Class 19
• Wed 09: Video tutorship (via Meet)
• Thr 10: Video tutorship (via Meet)
• Fri 11: Video meeting 11 (via Meet)
Periodo, 14/12 - 18/12: Winter Break
Periodo, 21/12 - 25/12: Winter Break
Periodo, 28/12 - 01/01: Winter Break
Week 12, 04/01 - 08/01: Matrices and Determinants
• Mon 04: Video Class 20
• Tue 05: Video Class 21
• Wed 06: Video tutorship (via Meet)
• Thr 07: Video tutorship (via Meet)
• Fri 08: Video meeting 12 (via Meet); Submit Assignment 4 (via Homework/Classroom)
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8. Advanced Algebra I
Week 13, 11/01 - 15/01: Linear Equation Systems
• Mon 11: Video Class 22
• Tue 12: Video Class 23
• Wed 13: Midterm 4, proctored on Meet, upload to Homework
• Thr 14: Video tutorship; review of Midterm 4
• Fri 15: Video meeting 13 (via Meet)
Week 14, 18/01 - 22/01: Linear Equation Systems
• Mon 18: Video Class 24
• Tue 19: Video Class 25
• Wed 20: Video tutorship (via Meet)
• Thr 21: Video tutorship (via Meet)
• Fri 22: Video meeting 14 (via Meet)
Week 15, 25/01 - 29/01: Linear Equation Systems
• Mon 25: Video Class 26
• Tue 26: Video Class 27
• Wed 27: Video tutorship (via Meet)
• Thr 29: Video tutorship (via Meet)
• Fri 30: Video meeting 15 (via Meet)
Week 16, 01/02 - 05/02:
• Tue 02: Submit Home–exam, (via Homework/Classroom)
• Thr 04: Final Exam
Week 17, 08/02 - 12/02: Final Exam
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