Learning outcome

A candidate who has passed the course will have a learning outcome in the form of acquired knowledge, skills, and general competence, as described below.

Knowledge

The candidate will:

• Understand how to form linear equations into matrix form
• Understand how linear equations can pose over determined, under determined, and degenerate problems, how this analysis is related to the matrices and vectors of the formulation, and how these cases can be handled
• Have an overview of relevant methods for matrix decomposition
• Have knowledge about how linear equations show up in various scientific problems
• Have a basic knowledge of the main numeric concepts related to solving linear equations

Skills

The candidate will:

• Be able to form linear equations in scientific problems into matrix form
• Be able to analyse linear equations with regards to solvability, and how to treat under determined and over determined problems
• Be able to use matrix decomposition methods to solve scientific problems using computers
• Be able to solve relevant scientific problems by posing them in a linear algebra form
• Be able to understand relevant computational methods in a linear algebra perspective

General competence

The candidate will:

• Understand how concepts from linear algebra have a wider use
• Understand scientific precision, and the role of theorems and proofs
• Get experience in presenting scientific theory and applications

Course Description

• Brief repetition: Linear equations, row equivalence, rank, Gauss elimination, Echelon form, LU-decomposition.
• Vector spaces, linear mappings, basis, coordinates and coordinate changes.
• Orthogonality, inner products, QR-decomposition. Least squares method.
• Determinants. Eigenvalues, similar matrices, diagonalization. (Jordan canonical form). Functions of matrices, Cayley-Hamilton's theorem. Orthogonal projections. The spectral theorem for normal matrices. Quadratic forms. Main axis theorem. Sylvester theorem. Singular value decomposition.
• Key applications of linear algebra such as: the Newton method and iterations in nonlinear equations and in optimization problems. Solution of linear differential equations, stability. Controllability in systems theory. Reaction invariance in models of reacting systems. Dimensionless variables in scientific models. Linear regression models.

Teaching and Learning Methods

The course content is to be presented by lecturer(s) and PhD students in a seminar form with variations, depending on the number of students. The students will be expected to work on case studies which are to be solved using computational tools.

Assessment Methods

The final test will count 100% in the course grade.

Minor adjustments may occur during the academic year, subject to the decision of the Dean