Syllabus
Elementary Linear Algebra and Analytical Geometry MAT 105, Fall 2010 AUBG
MAT 105a: MW 12:30-13:45, BAC 203; Mat105b 17:45-19 BAC 202
Instructor: Tatiana Ivanova, Assoc. Professor of Mathematics at AUBG,
Senior Researcher at the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Office and telephone: R 317, NAB, ext 491 tatyana@aubg.bg
Office hours: during the Problem solving sessions, Mo 19:15-20:30, R 201 NAB and by appointment.
Each week there will be
1. Problem solving session /Office hours given by the Instructor
Mo 19:15-20:30 R 201 NAB
and
2. Tutorial session given by Yasen Kolev – (excellent Math major student)
We – 19:15- 20:15 R201 NAB
This will complement the session given by the lecturer.
Course Description.
The course offers a general view to some important ideas and techniques in the field. Starting with a
discussion of systems of linear equations (the natural source of the subject) the important technique of
matrices, matrix operations and determinants is considered. An illustration of the general concepts in
the plane and space geometry helps the students to cultivate their intuition and interpretative skills in
the area. An elementary introduction to General vector spaces, Linear Transformations and Eigenvalue
problems initiates the students into this powerful technique.
Prerequisite: completion of Introductory Mathematics requirement.
Gen Ed: quantitative analysis. Cr. 3
Literature:
1) David C.Lay, Linear Algebra and its Applications , 3rd edition (Basic Textbook)
2) Gilbert Strang, Linear Algebra and its Applications , 3rd edition (Suplementary material, can be
found on Reserve in the AUBG Library. This book is excellently structured , but more
challenging, and contains a greater variety of problems, which are not routine )
Course Summary:
I. Systems of linear equations, Matrices and Gaussian elimination: 4 lectures
[1] Chapter 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8*, 1.9*
 Systems of linear equations [1, 1.1 ]
 Gaussian elimination -Row reductions and echelon forms [1, 1.2]
 Vectors in R^n, linear combinations, Span {v_1, …, v_k} [1, 1.3]
 The matrix equation Ax=b [1, 1.4]
 Solution sets of linear systems [1, 1.5]
 Linear independence [1, 1.7]*
 Linear Transformations. The matrix of a linear transformation [1, 1.8, 1.9] *
II. Matrix Algebra
4 lectures
[1] Chapter 2, 2.1, 2.2, 2.3, 2.8, 2.9,
 Matrix operations [2, 2.1]
 The inverse of a matrix. Elementary matrices [2, 2.2]



Characterization of invertible matrices [2, 2.3]
Subspaces of R^n. Column space and nullspace of a matrix. [2, 2.8]
Dimension and rank, the rank theorem, the basis theorem. Rank and the invertible matrix
theorem.* [2, 2.9]
III. Vector spaces
5 lectures
[1], Chapter 4, 4.1, 4.2, 4.3, 4.5 , 4.6, 4.4*, 4.7*
 Vector spaces and subspaces [4, 4.1]
 The Four Fundamental Subspaces of a matrix [4, 4.2]
 Linearly independent sets, bases, and dimension [4, 4.3]
 The dimension of a vector space [4, 4.5]
 Rank (The Fundamental Theorem of Linear Algebra, Part I) [4, 4.6]
IV. Determinants: 3-4 lectures
[1], Chapter 3.1, 3.2, 3.3, 2.3, 2.8, 2.9
 Introduction to determinants,
 Properties of the Determinant
 Applications of determinants : Cramer’s rule, volume.
V. Eigenvalues and eigenvectors 4-5 lectures
[1], Chapter 5, 5.1, 5.2, 5.3, 5.4, Chapter 7, 7.1
 Eigenvectors and eigenvalues [5, 5.1]
 The Characteristic equationc [5, 5.2]
 Diagonalization, diagonalizing matrices [5, 5.3]
 Diagonalization of symmetric matrices * [7, 7.1]
 Eigenvectors and linear transformation * *[5, 5.4]
VI Orthogonality:
[1], Chapter 6, 6.1, 6.2, 6.3, 6.4 2 lectures ,
 Inner product, length and orthogonality. Orthogonal sets
 Lines and planes in 3 –space
 Orthogonal bases*
Assessment: Your grade will be formed by
final exam
40%of the final grade ( maximum of 100 points)
two quizzes (midterms)
each
giving 25% of the final grade (each gives a
maximum of 100 points)
4 or 5 popquizzes, participation in class 10% of the final grade
Attendance: you can add to the total grade maximum 5 points (your presence in each class
gives 0.25 points) .
Total = 40% F +25%M1+ 25%M2 + Accumulated points for Popquizzes
+ Attendance(Max 5 points)
Max total
105
Points/Grade Map:
A 100-105
A- 94-99
B+ 88-93
B 82-87
B- 76-81
C+ 69-75
C 62-68
C- 55-61
D+ 48-54
D 40-47.
Every class session I will ask the students to participate actively in the solutions of the problems.
Problems from the home assignments will be often discussed in class and in the problem solving sessions.
Solving them is a good preparation for popquizzes, quizzes and final.
• The popquizzes will be scheduled sporadically (typically at the end of the class) and will last 15-20
minutes. (There will be two popquizzes in the period 27.01- 8.02 ). Each popquiz will contain 2-3 routine
problems and will give in total 2, 3, or 4 points. The grade for the popquizzes is cumulative, based on the
total number of points received for (all) popquizzes. The maximal total number is 16, but only 10 points
participate in the final grade. The problems are designed so, that computation should not create difficulties,
and only solutions which are with no error will bring points. There is no make up for popquizzes.
• Each quiz will hold 70 min and will consist of 5 or more problems. Some will be routine (up to 90
points). Do expect also one or two nontrivial problems (given as bonus problems). Each exam (quizzes and
final) is designed for 130 points max, (30 are bonus) but only 100 points participate in the final grade.
• There will a be make up for each quiz on a date and time fixed by the professor.
• The final will be comprehensive, i.e., over all the material covered by the course.
Attendance and participation in class:
Student will be expected to participate actively in class. Attendance will be checked regularly, starting from
the first week of the semester.
Student with 3 or more non-excused absences will be kindly advised to withdraw from the course. An
excuse should be asked from the professor before the class.
In case of absences due to medical reasons, the students are kindly requested to inform the professor via email (before the class). It is a responsibility of the students to take care and prepare with the material
covered during their absence. I expect you to come to class prepared (with written home work, and having
read the assigned text if there is such) and to participate actively during the class.
Important dates:
• Midterm 1: October 6, 19:30-21, Auditorium, Main Building
• Make up M1: November 8, 19:15-20:40 pm, Auditorium MB
• Midterm 2: November 24, 19:30-21, Auditorium, Main Building
• Make up M2 December 1, 19:15-20:40 (this date can be changed)
• Popquizzes (4-5) will be scheduled sporadically and will last 15 minutes, at the end of the class . There
is no make up for popquizzes. Approximate dates for the first two are: P1- September 13; P2-September
27.
Academic Honesty Policy. Students are expected to adhere to the Academic Honesty Policy stated in the
Catalog and the Student Handbook. Consequences for violations are included in the Catalog and the Student
Handbook.
Exam policies. During quizzes, exams and the final all that you will need and will be allowed to use is a
pen/pencil and the sheets of paper I give you. (all mobiles switched off , no textbooks, notes, calculators,
etc.). You should work strictly by yourself – you should not communicate in any way with your
classmates – violation of this will be considered cheating with all the ensuing consequences (see the AUBG
documentation for the consequences of cheating). Cheating is not only talking to the person next to you
(talking about anything: math, the problems, the weather, last nights party…) but also intentionally making
your work available to others during the exam.
Assignments: Often I will assign sections from the textbook for you to read ahead and from time to time I
will make reading quizzes to check if you have read the assigned part. I will also expect that on the average
you spend 3-4 hours per week (at least) on top of the regular Linear Algebra classes working on problems
from the book, or additional problems given in class. I will regularly give a list of problems as optional
homework, i.e., I will not collect these optional homeworks but you are strongly encouraged to do as many
as possible.
Office hours: Mo, 19:15-20:30, R201, NAB during the problem solving sessions. If the “official” office
hours above are not convenient for you, please, contact me to arrange some other time. Don’t be shy to ask.
There are no stupid questions.
Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any
changes. The above distribution of weeks per chapter is only approximate.