Course Syllabus

Draft Syllabus

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Course Description

Linear algebra is a branch of mathematics that focuses on linear equations, linear transformations, vector spaces and matrices. Its far-reaching impact extends across disciplines like physics, engineering, computer science, and economics, with practical applications spanning computer graphics, machine learning, signal processing, economic models and beyond.

This course will cover fundamental concepts of linear algebra from a mathematical perspective, including systems of linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality, least squares, and symmetric matrices. It has a hands-on format where a major focus is placed on mathematical application of concepts and problem solving. In addition, students will have the opportunity to participate in field studies to learn how professionals from academia and industry utilize linear algebra in their work.

The course will cover the following modules (modules 6-7 will be covered if time allows for it):

1. Linear equations 
- Systems of linear equations
- Row reduction and echelon forms
- Vector equations
- Matrix equation 
- Linear systems and linear independence
- Linear transformations

2. Matrix algebra
- Matrix operations
- The inverse of a matrix
- Partitioned matrices
- Matrix factorizations
- Subspaces of Rn
- Dimension and rank

3. Determinants
- Introduction to determinants
- Properties of determinants
- Cramer’s rule, volume, and linear transformations

4. Vector spaces
- Vector spaces and subspaces
- Null spaces, column spaces, row spaces, and linear transformations
- Linearly independent sets
- Coordinate systems
- The dimension of a vector space

5. Eigenvalues and eigenvectors
- Eigenvectors and eigenvalues
- The characteristic equation
- Diagonalization
- Eigenvectors and linear transformations
- Complex eigenvalues

6. Orthogonality and least squares
- Inner product, length, and orthogonality
- Orthogonal sets
- Orthogonal projections
- The Gram–Schmidt process
- Least-squares problems
- Machine learning and linear models

7. Symmetric matrices and quadratic forms
- Diagonalization of symmetric matrices
- Quadratic forms
- Constrained optimization
- Singular value decomposition

Learning Objectives

By the end of this course, students will be able to:

- Apply fundamental concepts of linear algebra during mathematical reasoning
- Identify and provide examples of expressions that include systems of linear equations, vectors and matrices
- Solve systems of linear equations using various methods 
- Perform matrix operations 
- Analyze mathematical expressions and describe solutions of systems
- Develop the ability to communicate mathematical ideas clearly and effectively, using appropriate mathematical notation and language.

Faculty

TBA

Readings

Field Studies

We will have two course-integrated field studies to learn how linear algebra is utilized in academic research and industry work.

Field studies may include:

Visit to laboratories at Kungliga Tekniska Högskola (Royal Institute of Technology), Stockholm, Sweden

Visit to companies utilizing linear algebra for engineering applications or machine learning

Guest Lecturers

Guest lecturers may be invited to talk about topics of particular interest to students.

Approach to Teaching

The course consists of interactive lectures, class exercises, and problem-solving. Discussion of applications is included, especially during field studies.

DIS Accommodations Statement 

Your learning experience in this class is important to me.  If you have approved academic accommodations with DIS, please make sure I receive your DIS accommodations letter within two weeks from the start of classes. If you can think of other ways I can support your learning, please don't hesitate to talk to me. If you have any further questions about your academic accommodations, contact Academic Support acadsupp@dis.dk. 

Expectations of the Students

• You should participate actively during lectures, discussions, group work, and exercises.
• Laptops may be used for note‐taking, fact‐checking, or assignments in the classroom, but only when indicated by the instructor. At all other times, laptops and electronic devices should be put away during class meetings.
• Readings must be done prior to the class session. 
• In addition to completing all assignments and exams, you need to be present, arrive on time, and actively participate in all classes and field studies to receive full credit. Your final grade will be affected, adversely, by unexcused absences and lack of participation. Your participation grade will be reduced by 10 points (over 100) for every unexcused absence. Remember to be in class on time!
• Classroom etiquette includes being respectful of other opinions, listening to others and entering a dialogue in a constructive manner.
• You are expected to ask relevant questions in regards to the material covered.
• Excuses for any emergency absences must be given beforehand. It is the responsibility of the student to make up any missed coursework.

Evaluation

To be eligible for a passing grade in this class, all of the assigned work must be completed.

You are expected to turn in all assignments on the due date. If an assignment is turned in after the due date, your assignment grade will be reduced by 10 points (over 100) for each day the submission is late. 

Grading

Active participation: Includes attendance, preparation for lectures and other sessions, active participation in learning activities, class discussions, group work and problem solving. It also includes active participation during field studies and presentation of reflections on how they are related and relevant within the context of the course.

Assignments:  Homework problems will be posted on Canvas. Students are allowed to work in groups to complete their homework. A pdf scan or clear photo of their work must be submitted on Canvas. Problems from MyMathLab will also be assigned. Late submissions of homework will be deducted by 10 points for each day the homework is late.

Short tests: Tests that cover specific modules of the course will be posted on Canvas and be open for 4 hours. You are expected to work on your own. Once you start taking a quiz, you will have 30 min to complete it. You should provide final answers of questions on Canvas, and upload a pdf scan or clear photo of your work.

Exam: At the end of the semester, you will take an exam that covers all topics from the course. The exam will  be open for 24 hours. You are expected to work on your own. Once you start taking the exam, you will have 2 hours to complete it. You should provide final answers of problems on Canvas, and upload a pdf scan or clear photo of your work. 

Academic Regulations

Please make sure to read the Academic Regulations on the DIS website. There you will find regulations on:

• Course Enrollment and Grading
• Academic Expectations and Honor Code

DIS - Study Abroad in Scandinavia - www.DISabroad.org