Applied Linear Algebra - 2026
Repository for ALI1-S26 at VIA
Course information¶
- Course responsible: Associate Professor Richard Brooks, rib@via.dk
- 5 ECTS (European Credit Transfer System), corresponding to 130 hours of work
- 11 sessions, each with a duration of 4 lessons + 1 session for the exam
- Bachelor level course
- Grade: 7-step scale
- Type of assessment: 4-hour written exam (see exam description in the menu at the top)
Prerequisites¶
It is important that you recap some of your high-school math. Most importantly:
- Linear equations
- Systems of linear equations
- Vectors and vector operations
- Differential equations
Lectures and course organization¶
The course is scheduled to start Friday February 6 and will continue up until and including May 1. All sessions are from 8:20 to 11:50 in room C05.15. In general, each session is made up of four activities:
- At the beginning of each session, there will be a short recap of the previous session.
- We then go through the exercises from the previous session.
- We will go through the theory of the current session.
- After classes, and before the next session, you will have to solve exercises from the current session.
This then loops back to (1) at the beginning of the next session.
There are no mandatory assignments, but it is highly recommended to work on the exercises for each session. No instruction is provided for the exercises so you will have to work on them on your own or form study groups.
Course content and learning objectives¶
Applied Linear Algebra focuses on understanding and applying the core concepts of linear algebra to solve real-world problems. The course explores vector spaces, matrices, eigenvalues, and eigenvectors, emphasising their practical applications in fields such as computer graphics, machine learning, and engineering. The course is designed to provide students with a solid foundation in linear algebra, enabling them to tackle complex problems and develop analytical skills.
Learning Objectives
- Linear Systems: Understand the concept of a linear system of equations, how to represent them with matrices, and how to solve them using row reduction.
- Matrix Algebra: Perform matrix operations including addition, multiplication, and inversion. Understand matrices as transformations and systems of linear equations, and learn to use matrices for practical problem solving.
- Determinants and Invertibility: Compute determinants of matrices and understand their geometric and algebraic significance. Use determinants to assess matrix invertibility and to solve linear systems.
- Linear Transformations: Understand the concept of linear transformations. Learn to represent linear transformations with matrices and how to compose linear transformations.
- Vectors and Vector Spaces: Understand the fundamental concepts of vectors, vector operations, and vector spaces. Learn to interpret vectors algebraically and geometrically, and reason about spans, bases, dimensions, and linear independence.
- Coordinate Mappings and Change of Basis: Understand the concept of coordinate mappings and change of basis. Learn to represent vectors in different bases and how to change the basis of a vector space.
- Eigen-Basics: Understand the concept of eigenvalues and eigenvectors. Learn to find eigenvalues and eigenvectors of a matrix and how to diagonalize a matrix.
- Systems of Differential Equations: Apply linear algebra techniques to solve systems of first-order differential equations. Understand the connection between eigenvalues, eigenvectors, and the behaviour of dynamic systems.
- Orthogonality and Least Squares: Explore orthogonality in vector spaces, apply the Gram-Schmidt process, and solve least squares problems. Understand projections and their role in approximating inconsistent systems and fitting models to data.
- Symmetric Matrices, SVD and PCA: Analyse symmetric matrices, perform Singular Value Decomposition (SVD), and understand its application to data reduction and Principal Component Analysis (PCA).
Changes from 2025
- Added more emphasis on Linear Transformations.
- Added Coordinate Mappings and Change of Basis.
- Added explicit reference to PCA
These topics were not present in previous exam cases but are now included in the exam.
Resources¶
Lay: Lay, David C. Linear Algebra and its applications, 4th edition. (e-book, up to students to retrieve a copy). All chapters and exercises referenced will be to the 4th edition. Make sure you have the correct edition or else the exercise numbers will not match.
Note, for each lesson I have uploaded the presentations that accompany the book. I will in no way use these during classes and they are merely uploaded for your notes/convenience. The notes that I have uploaded are an electronic version of my personal lecture notes and contain most of the (relevant) material for the sections in questions. Since some of the exercises are from the book, you may see [M] associated with exercises. This means that you are supposed to solve the exercise using Matlab. Disregard this and use Python.
Non-session specific resources such as the exercises from the book, solutions, old exam cases, etc. can be found her:
This folder is always accessible in the menu at the top.
The Wiseflow code for all flows that are used during the course is always 0000. This is not the code for the actual exam in June, though.
Suggested online resources can be found in the menu at the top. These are not mandatory, but they can be useful for some students.
Make sure you install a working version of Jupyter Notebook and Python version 3.7 or higher. You can choose whichever IDE you want to work in as long as it can handle Jupyter Notebooks. Installing VS Code with a Jupyter Notebook extension seems to be a popular choice.
Historical Notes¶
Applied Linear Algebra was first offered in 2014 and is scheduled to be taught 1–2 times per year. The course responsible is Richard Brooks (RIB) who has been the only lecturer teaching the course.
| Grade | Count |
|---|---|
| 12 | 8 |
| 10 | 5 |
| 7 | 3 |
| 4 | 4 |
| 02 | 2 |
| 00 | 8 |
| -3 | 0 |