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 
In programs
TKMAS MECHANICAL ENGINEERING, Year 1 (compulsory)
Examiner:
Vincent Desmaris
Go to Course Homepage
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 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.
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 concept geometric vector in 3space
 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 (Gausselimination)
 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 LUfactorization 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 cofactorexpansion 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 leastsquares 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
Content
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, LUfactorization, iterative methods. 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
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.