Linear Algebra and Group Theory

The course is taken during the second semester of the Mathematics program, but can also be taken as a stand-alone course. The course can be included in the following programs: Mathematics program (N1MAT).

General entry requirements and the equivalent of the courses Foundations of Matematics MMG220 and Linear Algebra MMG250.

Abstract vector spaces over R and C. Subspaces and quotient spaces. Kernels and images. Isomorphism theorem. Dual spaces. Eigenvalues ​​and eigenvectors. Diagonalization. Generalized eigenspaces. Triangulation. Inner products and inner product spaces over R and C. Orthogonalization. Gram-Schmidt. Spectral theorem for normal operators and special cases of this. Jordan normal form (without proof). Cayley-Hamilton theorem. Minimal polynomials. Groups and homomorphisms. Abelian groups. Permutations and matrix groups. Side classes. Lagrange's theorem. Normal subgroups. Quotient groups. Kernels. Isomorphism theorems. Group action. Orbits and stabilizers. Classification of finite abelian groups (without proof).

After passing the course, the student should be able to:

• handle abstract vector spaces, including subspaces and quotient spaces,
• handle linear mappings and their matrices, and derive the kernel and image of a linear mapping,
• calculate eigenvalues ​​and eigenvectors and perform diagonalization,
• handle scalar products and scalar product spaces, with associated theory, including orthogonal projection and orthogonalization,
• define, exemplify and use the concepts of group and homomorphism,
• define and use the following concepts in group theory: cosets, normal subgroups, quotient groups, kernels, group action and stabilizers,
• prove and use important theorems in the course and carry out simpler proofs on their own in linear algebra and group theory.

The examination consists of a written exam at the end of the course. During the course, there may be optional assignments that give bonus points on the exam. Examples of such assignments are small written tests, labs, and oral or written presentations. Information about this is found on the course home page.

If a student who has been failed twice for the same examination element wishes to change examiner before the next examination session, such a request is to be granted unless there are specific reasons to the contrary (Chapter 6 Section 22 HF).

If a student has received a certificate of disability study support from the University of Gothenburg with a recommendation of adapted examination and/or adapted forms of assessment, an examiner may decide, if this is consistent with the course’s intended learning outcomes and provided that no unreasonable resources would be needed, to grant the student adapted examination and/or adapted forms of assessment.

If a course has been discontinued or undergone major changes, the student must be offered at least two examination sessions in addition to ordinary examination sessions. These sessions are to be spread over a period of at least one year but no more than two years after the course has been discontinued/changed. The same applies to placement and internship (VFU) except that this is restricted to only one further examination session.

If a student has been notified that they fulfil the requirements for being a student at Riksidrottsuniversitetet (RIU student), to combine elite sports activities with studies, the examiner is entitled to decide on adaptation of examinations if this is done in accordance with the Local rules regarding RIU students at the University of Gothenburg.

The grading scale comprises: Pass with Distinction (VG), Pass (G) and Fail (U).

The course is evaluated with an anonymous questionnaire and/or a discussion with the student representatives. The results of and possible changes to the course will be shared with students who participated in the evaluation and students who are starting the course.

The course content partially overlaps with the previous courses Linear Algebra II MMG400 and Algebraic Structures MMG500.