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MA20217: Algebra 2B

Follow this link for further information on academic years Academic Year: 2019/0
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Intermediate (FHEQ level 5)
Further information on teaching periods Period:
Semester 2
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this module you must take MA10209 AND take MA10210 AND take MA20216
Further information on descriptions 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 Cayley-Hamilton 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.
Further information on programme availabilityProgramme availability:

MA20217 is Compulsory on the following programmes:

Department of Computer Science
  • USCM-AFB20 : BSc(Hons) Computer Science and Mathematics (Year 2)
  • USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 2)
  • USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 2)
  • USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 2)
  • USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 2)
  • USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 2)
Department of Mathematical Sciences
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 2)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 2)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 2)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 2)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 2)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 2)

MA20217 is Optional on the following programmes:

Department of Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 2)
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 2)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 2)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 2)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 2)
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 2)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 2)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 2)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 2)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 2)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AFB05 : BSc(Hons) Statistics (Year 2)
  • USMA-AFB05 : BSc(Hons) Statistics (Year 3)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 2)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 2)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
Department of Physics
  • USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
  • USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
  • USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
  • USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 3)
  • USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
  • USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)

Notes: