Department of Mathematical Sciences, Unit Catalogue 2010/11 |
MA20217: Algebra 2B |
 Credits:
| 6 |
| Level:
| Intermediate |
| Period: | This unit is available in... |
| Semester 2 |
| Assessment:
| EX100 |
| Supplementary Assessment: | Supplementary assessment information not currently available (this will be added shortly) |
| Requisites:
| Before taking this unit you must take MA10209 and take MA10210 and take MA20216 or equivalent units from MA10001 - MA10006. |
| 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. |