Subject Outline for 620-222

Please note that this outline is tentative; individual topics may not be achieved on time, or at all.
LIN. ALG. Fields 
29 July Definition and examples, subfields 
30 July The integers modulo a prime, algebraically closed fields 
Revision
31 July Vector spaces and subspaces 
5 Aug Spanning, linear independence and bases
6 Aug Dimension, sums and intersections of subspaces
7 Aug Linear transformations and change of basis
Canonical forms
12 Aug Invariant subspaces
13 Aug Minimal polynomials
14 Aug More on minimal polynomials
19 Aug Triangular form and the Cayley-Hamilton Theorem
20 Aug Jordan canonical form Ass. 1 out
Inner product spaces
21 Aug Complex inner products
26 Aug More on inner products; orthogonal complements  
27 Aug Adjoints, Hermitian and normal matrices, isometries Ass. 1 due
The Spectral Theorem
28 Aug The spectral theorem
2 Sep Polar form
GROUPS Definition and Examples of Groups
3 Sep General discussion, definition and examples  
4 Sep More examples: matrix and permutation groups
9 Sep Mid-semester Test M-S test
Subgroups and isomorphism
10 Sep Subgroups
11 Sep Cyclic subgroups
16 Sep Isomorphism, products
Lagrange's Theorem
17 Sep Cosets, Lagrange's Theorem and its consequences
 Quotient groups and Homomorphisms
18 Sep Normal subgroups, Quotient groups
Mid-Semester Break M-S break
7 Oct Homomorphisms, The Isomorphism Theorem
 Group Actions
8 Oct Definition and examples
9 Oct Orbit-Stabiliser relation
Groups acting on themselves
14 Oct Two actions of groups on themselves 
15 Oct Conjugacy  Ass. 2 out
16 Oct Applications to group theory 
Euclidean Isometries
21 Oct Definition and basic properties 
22 Oct Classifying isometries Ass. 2 due
23 Oct Compositions and conjugacy classes  
Wallpaper groups
28 Oct Finite symmetry groups  
29 Oct Lattices and point groups
30 Oct Classification of wallpaper groups