Module description
Syllabus:
Inner product spaces: inner products for real and complex vector spaces, orthogonal and orthonormal spaces, projections. Special classes of matrices, including unitary and orthogonal matrices. Spectral theory, including the Cayley-Hamilton theorem, eigenspaces and multiplicities, Jordan normal form. Quadratic forms: definitions, diagonalisation and geometric applications.
Assessment details
Written examination and class tests.
Exercises or quizzes will be set each week to be handed in the following week. These problems will be discussed in the tutorials and solutions will be available.
Educational aims & objectives
The module builds on the ideas and tools introduced in Linear Algebra and Geometry I. It systematically covers the concepts of eigenvalues and eigenvectors of nxn matrices, diagonalisation of matrices and the geometric interpretation of these ideas.
Learning outcomes
Understand the definition of inner products on vector spaces and be able to find orthogonal or orthonormal bases for an inner product space. Understand and be able to apply basic results about the spectral theory of nxn matrices, including the Cayley-Hamilton theorem and Jordan normal form. Understand the definition and properties of quadratic forms (including diagonalisation) and apply these concepts in a geometric context to study simple examples of quadrics (including conic sections).
Teaching pattern
Three hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists