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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
TMV166 - Linear algebra and systems of linear equations  
Linjär algebra och system av linjära ekvationer
Syllabus adopted 2020-02-10 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55125
Open for exchange students: No
Only students with the course round in the programme plan

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0108 Examination 7,5c Grading: TH   7,5c   20 Mar 2021 pm J,  09 Jun 2021 pm J,  23 Aug 2021 pm J

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Vincent Desmaris

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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 above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Introductary course in mathematics and Programming in Matlab.


The purpose of the course is to, together with the other math courses in the program, provide a general knowledge in the mathematics required in further studies as well as in the future professional career.

Learning outcomes (after completion of the course the student should be able to)

  • account for the concept geometric vector in 3-space
  • know how to compute with vectors: addition and multiplication by scalar
  • understand and use scalar product and orthogonal projection
  • understand and the crossproduct and use it to compute area and volume
  • know the equations for the straight line and the plane
  • account for the concepts of matrices and vectors, and explain how these are used to write systems of linear equations
  • solve systems of linear equations by row reduction (Gauss-elimination)
  • determine if a vector is a linear combination of given vectors, and describe the linear span of a set of vectors
  • discuss geometrical properties of linear transformations and determine standard matrices for these, given sufficient information
  • determine the inverse of a matrix
  • use the theorem of invertible matrices in problem solving
  • determine a LU-factorization of a matrix when row interchanges are not required
  • define the concept of a subspace of Rn, and determine if a set of vectors is a subspace
  • define the concept of a basis for a subspace, determine the coordinates for a vector with respect to a given basis, and change between different bases
  • determine bases for the null space and column space of a matrix, and determine if a given vector belongs to either of these spaces
  • determine the rank of a matrix and use the rank theorem in problem solving
  • compute the determinant of a matrix of any dimension by cofactor-expansion or row reduction
  • utilize the basic properties of determinants in problem solving
  • determine the eigenvalues and eigenvectors of a matrix
  • diagonalize a matrix and use this in problem solving, for example to solve systems of ordinary differential equations
  • compute the inner product of two vectors, the norm of a vector, and the distance between two vectors
  • explain what is meant by an orthogonal basis for a subspace, and decompose a vector into two orthogonal components if given such a basis
  • use the least-squares method for model fitting
  • use the spectral theorem in problem solving
  • know how to solve linear systems of equations by LU factorisation and iterative methods


The course is about matrices and systems of linear equations.  Equal emphasis is put on the three pillars: mathematical theory, analytic techniques, and numerical computation.  Geometric vectors, scalar product, cross product.  Equations for the line and the plane.  Systems of linear equations, Gauss elimination. Matrix algebra, matrix inversion. Determinant. Vector spaces, the Euclidean vector space Rn, subspaces, linear independence, basis, dimension, coordinates, change of basis. Linear transformations: Matrix representation. Applications to rotations, reflections and projections. Transformations from Rn to Rm. Null space (kernel), column space (range), the rank (dimension) theorem. Orthogonality, orthonormal basis, orthogonal projection. The method of least squares. Eigenvalues, eigenvectors and diagonalization. Numerical solution of systems of linear equations: Matrix norms, conditioning numbers, LU-factorization, iterative methods. MATLAB applications in mechanics.


Instruction is given in lectures and classes. More detailed information will be given on the course web page before start of the course.


S. Larsson, A. Logg, A. Målqvist:  Analys och linjär algebra, del III:  Linjär algebra och system av linjära ekvationer

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course. 

Page manager Published: Thu 04 Feb 2021.