Linear Algebra

Overview

- Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension.
- Linear transformations, image, kernel and dimension formula.
- Matrix representation of linear maps, eigenvalues and eigenvectors of matrices.
- Matrix inversion, definition and computation of determinants, relation to area/volume.
- Change of basis, diagonalization, similarity transformations.
- Inner product spaces, orthogonality, Cauchy-Schwarz inequality.
- Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties.
- Basic computer application of linear algebra techniques.

Additional topics and applications, such as: Schur decomposition, orthogonal direct sums and geometry of orthogonal complements, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form.

Learning Objectives

It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of linear algebra; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

Skills

Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

Assessment

None