Academic Year:
 2012/3 
Owning Department/School:
 Department of Mathematical Sciences 
Credits:
 6 
Level:
 Intermediate (FHEQ level 5) 
Period: 
Semester 2

Assessment:
 EX 100% 
Supplementary Assessment:  MA20217 Mandatory extra work (where allowed by programme regulations) 
Requisites:
 Before taking this unit you must take MA10209 and take MA10210 and take MA20216 
Description:
 Aims: To introduce the students to basic abstract ring theory and provide a thorough structure theory of linear operators on a finite dimensional vector spaces.
Learning Outcomes: After taking this unit, students should be able to:
* Demonstrate understanding of the basic theory of rings.
* Factorise in various integral domains they have met throughout the course and demonstrate understanding of the general theory.
* State and prove the fundamental results on the structure theory of linear operators.
* Apply the structure theory of linear operators in examples. Determine characteristic polynomials, minimal polynomials, geometric and algebraic multiplicities as well as the Jordan normal form for a given linear operator. Calculate generalised eigenspaces.
Skills: Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).
Content: Elementary axiomatic theory of rings. Integral domains, fields, characteristic. Subrings and product of rings. Homomorphisms, ideals and quotient rings. Isomorphism theorems. Fields of fractions. Polynomial rings. Maximal ideals and prime ideals. Factorisation in integral domains. Unique factorisation in principal ideal domains. Eisenstein criterion and other criteria for factorisation in polynomial rings.
Revision of eigenvalues, eigenvectors and diagonalisability. Invariant subspaces and decomposition of linear operators . Minimal polynomials and the CayleyHamilton theorem. The primary decomposition theorem and generalised eigenspaces. Applications including calculations of powers and exponentials of matrices. Cyclic invariant subspaces. The Jordan normal form theorem. Applications.

Programme availability:

MA20217 is Compulsory on the following programmes:
Department of Computer Science
 USCMAFB20 : BSc (hons) Computer Science and Mathematics (Fulltime)  Year 2
 USCMAKB20 : BSc (hons) Computer Science and Mathematics (Fulltime with Thick Sandwich Placement)  Year 2
 USCMAAB20 : BSc (hons) Computer Science and Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USCMAFB13 : BSc (hons) Computer Science with Mathematics (Fulltime)  Year 2
 USCMAKB14 : BSc (hons) Computer Science with Mathematics (Fulltime with Thick Sandwich Placement)  Year 2
 USCMAAB14 : BSc (hons) Computer Science with Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USCMAFM14 : MComp (hons) Computer Science and Mathematics (Fulltime)  Year 2
 USCMAKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Fulltime with Thick Sandwich Placement)  Year 2
 USCMAAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
Department of Mathematical Sciences
 USMAAFB13 : BSc (hons) Mathematics (Fulltime)  Year 2
 USMAAKB14 : BSc (hons) Mathematics (Fulltime with Thick Sandwich Placement)  Year 2
 USMAAFB01 : BSc (hons) Mathematics and Statistics (Fulltime)  Year 2
 USMAAKB02 : BSc (hons) Mathematics and Statistics (Fulltime with Thick Sandwich Placement)  Year 2
 USMAAAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USMAAAB14 : BSc (hons) Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USMAAFM14 : MMath Mathematics (Fulltime)  Year 2
 USMAAAM15 : MMath Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
MA20217 is Optional on the following programmes:
Department of Mathematical Sciences
 USMAAFB15 : BSc (hons) Mathematical Sciences (Fulltime)  Year 2
 USMAAFB15 : BSc (hons) Mathematical Sciences (Fulltime)  Year 3
 USMAAKB16 : BSc (hons) Mathematical Sciences (Fulltime with Thick Sandwich Placement)  Year 2
 USMAAKB16 : BSc (hons) Mathematical Sciences (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USMAAAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFB05 : BSc (hons) Statistics (Fulltime)  Year 2
 USMAAFB05 : BSc (hons) Statistics (Fulltime)  Year 3
 USMAAKB06 : BSc (hons) Statistics (Fulltime with Thick Sandwich Placement)  Year 2
 USMAAKB06 : BSc (hons) Statistics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB06 : BSc (hons) Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 2
 USMAAAB06 : BSc (hons) Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
Department of Physics
 USXXAFB03 : BSc (hons) Mathematics and Physics (Fulltime)  Year 3
 USXXAKB04 : BSc (hons) Mathematics and Physics with Placement (Fulltime with Thick Sandwich Placement)  Year 4
 USXXAAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USXXAFM01 : MSci (hons) Mathematics and Physics (Fulltime)  Year 3
