620-222 Linear and Abstract Algebra

One of [01]620-112, 620-122, [05]620-194 or 620-211; a grade of H3 or better in the prerequisite is recommended. Students with a grade of H1 in 620-142 or [05]620-192 will be permitted to enrol on completion of additional reading.

This subject develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics which have important applications in science and technology. It also introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics and in theoretical physics.

Linear algebra topics include revision and extension of basic concepts; vector spaces including complex spaces, inner products, linear transformations, eigenvalues and eigenvectors, dual spaces and the connection with inner products; the spectral theorem for normal matrices; and Jordan normal form, without proof but with applications. Groups topics include abstract groups, examples including matrix groups and permutation groups; homomorphisms, normal subgroups, quotients and the first homomorphism theorem; group actions and permutation groups; and conjugacy classes and their interpretation in symmetry groups, permutation groups and matrix groups. Applications topics may include wallpaper groups, symmetry groups of regular polyhedra, and permutation groups.

Up to 24 pages of written assignments during the semester and a 45-minute written test held mid-semester (equally weighted, with a total of 0% or 20%); a 3-hour written examination in the examination period (80% or 100%). The relative weighting of the examination and the total assignment plus test mark will be chosen so as to maximise the student's final mark.