MA26610 Multilinear Algebra
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
It will continue the study of linear algebra, which was begun in Year 1, having benefited from students finishing Abstract Algebra (Algebra3 or Groups and Rings) in term 1.
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.
The module is divided in three part:
The first main topic is the Jordan canonical form and related results. Abstractly, this solves the classification problem for pairs (V, T) where V is a finite dimensional vector space over the complex numbers (or any other algebraically closed field) and T a linear selfmap of V, up to the equivalence relation induced by bijective linear selfmaps of V; more concretely, we classify n by n complex matrices A up to conjugation by invertible matrices P, i.e., the operation A > P^{1}AP.
Secondly, we treat bilinear, sesquilinear and quadratic forms on finite dimensional (real and complex) vector spaces. These structures are ubiquitous and fundamental in mathematics and many parts of the sciences. For example, the standard scalar product in R^n is an example. In passing we mention that the description of amplitudes, probabilities and expectation values in quantum theory places such structures at the very heart of how nature works at the smallest levels. We will cover orthonormal basis, GramSchmidt process, diagonalisation, singular value decomposition, hermitian forms and normal matrices, among other things.
The third part is concerned with a thorough discussion of the very useful concept of duality (dual vector spaces, dual linear maps, dual bases etc.) and its applications, and after that tensor, exterior and symmetric algebras and their basic properties.
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
Peter Lax, Linear Algebra and Its Applications, 2nd Edition, Wiley, 2007
View reading list on Talis Aspire
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.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (30%) 
Seminars  9 sessions of 1 hour (9%) 
Private study  61 hours (61%) 
Total  100 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D1
Weighting  Study time  

Assignments  15%  
Examination  85%  

Assessment group R1
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core optional for:

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Optional for:
 Year 3 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
This module is Option list A for:

UMAAG100 Undergraduate Mathematics (BSc)
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of GG14 Mathematics and Statistics
 Year 2 of GG14 Mathematics and Statistics
This module is Option list B for:

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 3 of GG14 Mathematics and Statistics
 Year 3 of GG14 Mathematics and Statistics

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
This module is Option list C for:
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics