Course syllabus - Linear Algebra

Objectives

Many problems in natural science and social science leads to large systems of linear equations. There exist different numerical methods to solve such systems with computers. Matrix is the mathematical key concept to analyze properties of systems of linear equations and the different numerical methods for solving the systems. The course aims to give basic understanding of matrices and their properties and this is achieved by studying vector spaces and linear transformations between such spaces. The theory of vector spaces is built up from axioms defining the abstract concept vector space. An essential goal is to improve the capability to grasp a mathematical theory and realize its usefulness in different applications. Furthermore the course should exercise the competence of writing and talking about mathematics in a correct and understandable way.

Learning outcomes

At the end of the course the student is expected to be able to:
-define what a vector space is and give examples of vector spaces.
-decide and argue if a given set with given operations is a vector space.
-find a base to a given vector space and find the relation between the coordinates of a vector in different bases.
-define what a linear transformation is and given a transformation find its matrix representation and explain, in geometrical terms, the theorems of linear transformations.
-find the relationship between the matrices of a linear transformation given in different bases.
-from a given base and a given scalar product construct an orthonormal base.
-calculate eigenvalues and eigenvectors and diagonalize linear transformations.
-write and talk in an understandable and correct way about the mathematics that is part of the course.

Course content

Block multiplication. The relationship between the solvability of systems of equations and the rank of the coefficient and the total matrix. Determinants: Cramer's rule, the adjoint and inverse matrices. Rn, general vector spaces and inner product spaces. Gram-Schmidt's method and projections in Rn. Linear maps,: Corresponding matrix, change of basis, dimension theorem. Eigen values and eigen vectors. Eigen space. Diagonalisation of matrices, the spectral theorem. Quadratic forms.

Tuition

Lectures and/or working in groups supported by the teacher.

Specific requirements

Algebra for Engineering 7,5 credits or equivalent.

Examination

Exercises (INL1), 3 credits, marks Pass (G)
Written and/or oral examination (TEN1), 4.5 credits, marks 3, 4 or 5

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass with credit, Pass, Fail