# MA266-10 Multilinear Algebra

Department
Warwick Mathematics Institute
Level
Dmitriy Rumynin
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### 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 (Algebra-3 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.

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

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.

## Study time

Type Required
Lectures 20 sessions of 1 hour (20%)
Online learning (independent) 9 sessions of 1 hour (9%)
Private study 13 hours (13%)
Assessment 58 hours (58%)
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 D
Weighting Study time
Assignments 15% 20 hours
Examination 85% 38 hours
##### Assessment group R
Weighting Study time
In-person 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 UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
• Year 2 of G103 Mathematics (MMath)
• Year 2 of G103 Mathematics (MMath)
• UMAA-GV17 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 UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Core optional for:

• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 2 of UMAA-G103 Undergraduate Mathematics (MMath)
• Year 2 of UMAA-G1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
• Year 2 of UMAA-G101 Undergraduate Mathematics with Intercalated Year

This module is Option list A for:

• Year 2 of UMAA-GL11 Undergraduate Mathematics and Economics
• Year 2 of UECA-GL12 Undergraduate Mathematics and Economics (with Intercalated Year)
• UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
• Year 2 of GF13 Mathematics and Physics
• Year 2 of GF13 Mathematics and Physics
• UPXA-FG31 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)
• USTA-GG14 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: