Introductory description

It is a second Linear Algebra module, where advanced linear algebra concepts are rigorously developed for students familiar with algebraic tools.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Liner maps: Jordan Normal form, Cayley-Hamilton theorem, primary decomposition, functions of matrices
Quadratic forms over R and C: orthonormal basis, Gram-Schmidt process, diagonalisation, singular value decomposition, hermitian forms and normal matrices
Tensors: tensor product of vector spaces as a quotient of the free vector space, universal mapping property, its basis, (n,k)-tensor on a vector space, change of basis
Further topics: dual space, dual linear map, bilinear forms, skew-symmetric forms, determinant, Darboux Theorem, Witt Extension Theorem, free associative algebra and tensor algebra, other algebras (exterior, symmetric, Clifford).

Learning outcomes

By the end of the module, students should be able to:

• develop full command of the theory and computation of the the Jordan canonical form of matrices and linear maps
• learn how to define and to compute functions of matrices
• develop the working knowledge of bilinear forms and quadratic forms
• master the concept of tensor and get proficient manipulating tensors

Indicative reading list

P M Cohn, Algebra, Vol. 1, Wiley, 1982
I N Herstein, Topics in Algebra, Wiley, 1975
Jörg Liesen and Volker Mehrmann, Linear Algebra, Springer, 2015
Peter Petersen, Linear Algebra , Springer, 2012
F. Gantmacher, The Theory of Matrices, American Mathematical Society, 2001

Subject specific skills

This module teaches students to carry out fundamental calculations with matrices, including the theory and computation of the Jordan canonical form of matrices and linear maps; bilinear forms, diagonalizing quadratic forms, and choosing canonical bases for these. After that the module introduces the notion of tensor, treating them rigorously.

Transferable skills

The algorithmic techniques taught have widespread "real world" applications. Examples include ranking in search engines, linear programming and optimisation, signal analysis, and graphics. To also include: clear and precise thinking; the ability to follow complex reasoning; constructing logical arguments, and exposing illogical ones; and formulating problems as algorithms, thereby enhancing understanding of details and rendering them suitable for computer implementation.