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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
TMV166 - Linear algebra  
Linjär algebra
Syllabus adopted 2019-02-21 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55127
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   21 Mar 2020 pm SB_MU   09 Jun 2020 pm J,  24 Aug 2020 pm J

In programs



Stig Larsson

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TMV165   Linear algebra


In order to be eligible for a first cycle course the applicant needs to fulfil the general and specific entry requirements of the programme(s) that has the course included in the study programme.

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 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 LU factorization
  • 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


Matrix algebra, matrix inversion and systems of linear equations. Determinants, the rank of a matrix and systems of linear equations. 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. Numerical solution of systems of linear equations: Matrix norms, conditioning numbers, LU-factorization. The least-squares method. Eigenvalues, eigenvectors and diagonalization. Orthogonality and orthonormal bases. 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.


Literature will be announced on the course web page before start of the course.

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course.
Examples of assessments are:
-selected exercises are to be presented to the teacher orally or in writing during the course,
-other documentation of how the student's knowledge develops,
-project work, individually or in group,
-written or oral exam during and/or at the end of the course.
-problems/exercises are to be solved with a computer and presented in writing and/or at the computer.

Published: Wed 26 Feb 2020.