Department of Mathematical Sciences, Unit Catalogue 2010/11 |
MA10210: Algebra 1B |
 Credits:
| 6 |
| Level:
| Certificate |
| Period: | This unit is available in... |
| Semester 2 |
| Assessment:
| EX 100% |
| Supplementary Assessment: | MA10210 Mandatory extra work (where allowed by programme regulations) |
| Requisites:
| Before taking this unit you must take MA10209 and while taking this unit you must take MA10207 |
| Description:
| Aims: To introduce the abstract theories of groups and vector spaces and demonstrate their power through applications. Learning Outcomes: After taking this unit the students should be able to: * State and prove fundamental results of group theory and linear algebra. * Demonstrate competence applying abstract ideas in specific examples. * Compute determinants, eigenvalues and eigenvectors. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) Content: Axiomatic development of group theory. Subgroups, homomorphisms, kernel and image. Order of an element and of a group; cyclic groups. Isomorphism. Permutations: cycle notation, sign of a permutation. Actions, orbits and Cayley's theorem. Cosets, Lagrange's theorem and applications. Orbit-Stabilizer theorem and applications. Axiomatic development of vector spaces. Extending a linearly independent set to a basis. Dimension, rank, nullity. Rank-Nullity theorem. Determinants: definition, properties and computation. Adjugate and inverse formula. Invertibility of matrices. Cramer's rule. Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicities. Diagonalisation. |