Entry requirements

Linear Algebra II. Several Variable Calculus or Geometry and Analysis III.

Learning outcomes

In order to pass the course (grade 3) the student should be able to

• give an account of important concepts and definitions in the theory of linear spaces over arbitrary fields;
• exemplify and interpret important concepts in specific cases;
• formulate important results and theorems covered by the course;
• describe the main features of the proofs of important theorems;
• express problems from relevant areas of applications in a mathematical form suitable for further analysis;
• use the theory, methods and techniques of the course to solve mathematical problems;
• present mathematical arguments to others.

Content

Linear spaces over arbitrary fields, sums and direct sums of subspaces, the dimension formula, quotient spaces. tensor product. Linear transformations. Linear functionals, the dual space, dual bases. The canonical isomorphism between a linear space and its bidual. Forms: bilinear, Hermitian, symmetric, alternating, quadratic. Inner product spaces: unitary, Euclidean, orthogonal projection, the method of least squares. Linear operators: Hermitian, symmetric, unitary, orthogonal, normal, polynomial, the spectral theorem (complex and real), simultaneous diagonalisation, eigenspaces and generalised eigenspaces, the characteristic polynomial and the minimal polynomial, Jordan’s normal form (complex and real). Polar decomposition. Orientation about matrix groups: the general linear group, the orthogonal groups, the unitary group.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course and assignments given during the course.