Course syllabus adopted 2026-02-19 by Head of Programme (or corresponding).
Overview
- Swedish nameLinjär algebra och system av linjära ekvationer
- CodeTMV166
- Credits7.5 Credits
- OwnerTKMSK
- Education cycleFirst-cycle
- Main field of studyMathematics
- DepartmentPHYSICS AND ASTRONOMY
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 44122
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0108 Examination 7.5 c Grading: TH | 7.5 c |
In programmes
Examiner
- Karine le Bail
- Associate Professor, Onsala Space Observatory, Space, Earth and Environment
Eligibility
General entry requirements for bachelor's level (first cycle)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Specific entry requirements
The same as for the programme that owns the courseApplicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Course specific prerequisites
Aim
The aim of the course is to study the theory and applications of linear algebra, with a clear connection to analytical and numerical methods for solving systems of linear equations.The course develops the students ability to use both analytical and numerical methods, and to use programming in Python as a natural tool for modelling and computation. By combining theory, computation, and applications, the course strengthens both theoretical understanding and practical problem-solving skills, and provides a solid basis for further studies in mathematics and engineering.
The student is expected to have prior knowledge corresponding to introductory courses in mathematical analysis (differential calculus and integral calculus) and introductory programming in Python.
Learning outcomes (after completion of the course the student should be able to)
- Understand and use the concept of geometric vectors in two and three dimensions, perform basic vector algebra, and use concepts such as scalar product, orthogonality, projection, and cross product to describe and analyse lines and planes in different coordinate systems, and to compute area and volume.
- Formulate and analyse systems of linear equations in vector and matrix form, use Gaussian elimination to solve such systems, describe solution sets via parameterisation, and relate these to the concepts of linear independence and linear mappings.
- Explain and use basic matrix algebra and determinants, explain the relationship between determinant, invertibility, and solvability of systems of linear equations, and give an overview of the use of matrix factorisations in solving such systems.
- Formulate and use the concepts of vector space, subspace, basis, dimension, and rank; work with inner products, orthogonality, and projections (e.g. using the GramSchmidt process); and apply these concepts in formulating and solving least-squares problems and in interpreting function spaces and general inner product spaces.
- Understand and use the concepts of eigenvalue and eigenvector, determine whether a linear mapping or matrix is diagonalisable and interpret diagonalisation in terms of eigenbases, apply the spectral theorem for symmetric matrices, and relate positive definite quadratic forms and positive definite matrices to properties of the eigenvalues.
- Formulate and analyse numerical methods for solving linear systems and eigenvalue problems, use basic direct and iterative methods in simple situations, and give an overview of issues of condition number, stability, and preconditioning.
- Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
Content
Geometric vectors: Vector algebra; scalar product, orthogonality, and projection; coordinate systems; cross product; line and plane.
Systems of linear equations: Vector algebra; Gaussian elimination with pivoting; vector equations and matrix equations; solution sets and parameterisation; linear independence; linear functions (linear maps/transformations).
Matrices: Matrix algebra; inverse matrix; determinant; rules for determinants; matrix factorisations and their relation to the solution of systems of linear equations.
Linear spaces: Vector spaces and subspaces; bases and components; dimension and rank; inner product, orthogonality, and projection; GramSchmidt process; least-squares method (normal equations/projection); general inner product spaces; function spaces.
Eigenvalue problems: Eigenvalues and eigenvectors; diagonalisation and eigenbases; spectral theorem for symmetric matrices and orthogonal eigenbases; quadratic forms and diagonalisation; positive definite quadratic forms; positive definite matrices.
Numerical solution of systems of linear equations: LU factorisation; iterative solution methods (Jacobi, GaussSeidel, conjugate gradient); condition number and preconditioning; numerical solution of eigenvalue problems (power method, inverse power method).
Applications and programming: Python as a computational tool for linear algebra; implementation, testing, and validation of methods; applications to curve fitting by the least-squares method and to the solution of linear systems of ODEs via diagonalisation.
Organisation
Teaching consists of lectures and exercise classes in smaller groups. More detailed information is provided on the course web page before the start of the course.
Literature
S. Larsson, A. Logg, A. Målqvist, MATEMATISK ANALYS & LINJÄR ALGEBRA (III): Linjär algebra och system av linjära ekvationerExamination including compulsory elements
Assessment is by a written examination. Assignments or short quizzes that may give bonus points towards the examination may be included.The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers about disability study support.