Linear Algebra

Aim

To give a unified framework for geometrical and algebraic techniques, with applications in analysis, mechanics, numerical analysis, mathematical statistics, control theory, linear optimisation and other subjects. After completing the course the students will be able to read and understand the linear algebra which is used in other courses within the programme and the linear algebra which can be found in technical articles. In order to handle this it is necessary to be able to

• solve systems of linear equations and know of the structure of the set of solutions
• work with inner product and vector product for geometrical vectors
• perform calculations with matrices and determinants
• describe the concept of a vector space and perform calculations with vectors and coordinates
• describe the concept of a linear mapping, determine the matrix of a linear mapping and calculate the null space and the range
• project orthogonally on subspaces and use the method of least squares
• use a change of basis in order to solve problems
• formulate the spectral theorem and use it in order to solve systems of differential equations and systems of difference equations

Content

Linear systems of equations. Geometrical vectors, straight lines and planes. Matrices. Vector spaces. Euclidean spaces. Determinants. Linear mappings. Eigenvalues and eigenvectors. Symmetric mappings. Quadratic forms. Systems of differential equations.