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## Syllabus for

TMV166 - Linear algebra Linjär algebra

Owner: TKMAS
7,5 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES

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

TKMAS MECHANICAL ENGINEERING, Year 1 (compulsory)

#### Replaces

TMV165   Linear algebra

#### Eligibility:

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.

#### Aim

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

#### Content

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.

#### Organisation

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

#### Literature

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.