Syllabus for Linear Algebra II

Learning outcomes

On completion of the course the student shall be able to:

• be able to give an account of and use basic vector space concepts such as linear space, linear dependence, basis, dimension, linear transformation;
• be able to give an account of and use basic concepts in the theory of finite dimensional Euclidean spaces;
• be familiar with the concepts of eigenvalue, eigenspace and eigenvector and know how to compute these objects;
• know the spectral theorem for symmetric operators and know how to diagonalise quadratic forms in ON-bases;
• know how to solve a system of linear differential equations with constant coefficients;
• be able to formulate important results and theorems covered by the course;
• be able to use the theory, methods and techniques of the course to solve mathematical problems;

Content

Linear spaces: subspaces, linear span, linear dependence, basis, dimension, change of bases. Matrices: rank, column space and row space. Linear transformations: their matrix and its dependence on the bases, composition and inverse, range and nullspace, the dimension theorem. Euclidean spaces: inner product, the Cauchy-Schwarz inequality, orthogonality, ON-basis, orthogonalisation, orthogonal projection, isometry. Quadratic forms: diagonalisation. Spectral theory: eigenvalues, eigenvectors, eigenspaces, characteristic polynomial, diagonalisability, the spectral theorem, second degree surfaces. Systems of linear ordinary differential equations.

Instruction

Lectures and problem solving sessions. Test or written assignment.

Assessment

Written examination at the end of the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.