| Unit name | Linear Algebra 2 |
|---|---|
| Unit code | MATH21100 |
| Credit points | 20 |
| Level of study | I/5 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Misha Rudnev |
| Open unit status | Not open |
| Pre-requisites |
MATH 11002 and MATH 11003 |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Linear algebra with more rigour and depth than at Level 1. Elementary field theory, vector spaces over arbitrary fields, linear transformations and matrices. The determinant is defined rigorously, leading to characteristic and minimum polynomials of a matrix, and the Cayley-Hamilton theorem. Reducing subspaces, block-diagonal form for reducible transformations, the Jordan normal form over C. Bilinear maps and forms, symmetric and hermitian matrices, orthogonal diagonalisation, analysis of quadrics over R and C. Group theory of invertible linear maps of special types.
Aims
To give a rigorous account of vector spaces, their subspaces, and quotient spaces over arbitrary fields and linear maps between them and of real and complex inner-product spaces. The holy grail of the unit is the Jordan Normal Form theorem.
Syllabus
Introduction on groups, rings, fields and permutations (3 lectures).
Vector spaces, linear maps bases and dimension, direct sums and quotients, dual spaces, eigenvalues (8 lectures).
Matrices: rank, conjugacy, singularity, exterior algebras, determinants (7 lectures).
Polynomial rings. Cayley-Hamilton theorem. Spectral theorem. Jordan form (10 lectures).
Some of bilinear and quadratic forms. Hermitian, inner product and Normed spaces. (6 lectures).
Revision (2 lectures).
Course materials will be available at http://www.maths.bris.ac.uk/~maxmr/la2.html
Relation to Other Units
This unit develops the linear algebra material from first year Linear Algebra & Geometry, giving a more general and abstract treatment, using the central algebraic structures, such as groups, rings, and fields. This material is a central part of Pure Mathematics; it is a prerequisite for Measure & Integration (was Analysis 3), and is relevant to other Pure Mathematics units at levels 3 and 4.
After taking this unit, students should have gained a thorough understanding of vector spaces and the natural maps between them and an appreciation of some of their main pure mathematical properties.
Transferable Skills:
Assimilation of abstract ideas. Reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.
Lectures, problem classes, problems to be done by the students, and solutions to these problems.
The final assessment mark for Linear Algebra 2 is calculated from a STANDARD RUBRIC 2 ½ -hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Neither calculators nor note sheets are permitted in this examination.
Linear Algebra, a pure mathematical approach by Harvey E. Rose (Birkhauser Verlag, 2002)
Students will be given printed notes which largely follow parts of the book above.
