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Linear Algebra 2 (MATH 21100)

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Administrative Information

  1. Unit number and title: MATH 21100 Linear Algebra 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1992
  6. Unit Organiser: Misha Rudnev
  7. Lecturer: Drs C. Harris and M Rudnev
  8. Teaching block: 2
  9. Prerequisites: MATH 11005 Linear Algebra & Geometry

Unit 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. 

General Description of the Unit

In Level 1 students meet real vector spaces, matrices with real entries, and techniques for their manipulation. This unit is, by comparison, more general, and so more abstract, investigating vector spaces, linear transformations and matrices over an arbitrary field, and bilinear and quadratic "forms" over the real or complex numbers. The unit is "pure" in the sense that the emphasis is on insight rather than techniques, this insight being attained through careful use of definitions of the key concepts, and the formulation and proof of the key results. A common theme is the choice of a basis in order to reduce a particular problem to one more readily visualised.

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.

Teaching Methods

Lectures, problem classes, problems to be done by the students, and solutions to these problems.

Learning Objectives

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.

Assessment Methods

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.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

Transferable Skills

Assimilation of abstract ideas. Reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

Texts

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.

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