Message from Geo re his office location.
Course information and marking scheme
Problem Set 1 (due on Thursday January 22nd)
Problem Set 2 (due on Thursday January 29th)
Problem Set 3 (due on Thursday February 5th)
Problem Set 4 (due on Thursday February 12th)
Problem Set 5 (due on Thursday March 12th)
Problem Set 6 (due on Thursday March 19th)
Problem Set 7 (due on Thursday March 26th)
Problem Set 8 (due on Thursday April 2nd);
Problem Set 9 (not to be handed in).
Definitions and basic results related to groups
Notes on orthogonal projections
Partial solutions to Problem Set 1
Partial solutions to Problem Set 2
Partial solutions to Problem Set 3
Partial solutions to Problem Set 4
Correction to solution of question 5, problem set 4.
Review questions and info about term test.
Partial solutions for questions 1-3 on term test.
Solutions for questions 4 and 5 on term test.
Info about term test marks.
Partial solutions to Problem Set 5
Partial solutions to Problem Set 6
Partial solutions to Problem Set 7
Partial solutions to Problem Set 8 dual
space questions.
Partial solutions to group theory questions.
Material covered on Thursday Feb 26:
Minimal polynomials.
Material covered on March 2 and 3:
Minimal polynomials: examples.
Description of Jordan canonical form.
If g(t) and h(t) are polynomials with coefficients in F,
then g(T)h(T)=h(T)g(T) for all linear operators on V.
Generalized eigenspaces; cycles of generalized eigenvectors;
Theorem 7.1.
Material covered on March 9 and March 10:
The characteristic polynomial of the restriction of T
to the generalized eigenspace corresponding to eigenvalue
c is equal to (-1)^d (t-c)^d.
Theorem~7.2(b).
Example: Finding a Jordan canonical basis.
Theorem 7.6 and Corollary.
Material covered on March 16 and March 17:
Theorem 7.5, part a).
Idea of proof of Theorem 7.7.
Dot diagrams (p.498); Theorem 7.9.
Examples using Theorem 7.9 to compute Jordan canonical form.
Using Theorem 7.9 to determine when two particular linear operators
on V have the same Jordan canonical form.
Corollary on page 500.
Material covered on March 23 and March 24:
Similarity of matrices and Jordan canonical form;
Dual spaces - Section 2.6.
Definition of group and subgroup; examples; definition
of homomomorphism. (See notes above.)
Material covered on March 30 and April 1:
Example using the subgroup test; properties of homomorphisms.
Rotations and reflections - page 473 of [FIS] (Friedberg, Insel and Spence)
Theorem 6.23 (p.387 of [FIS]) (see also Theorem 6.45 and Corollary (p.474)).
Dihedral group as the group of symmetries of a regular n-gon.
Material covered on April 6 and 7:
Example: homomorphism from dihedral group of order
8 to the group of 2 by 2 invertible complex matrices.
Definition of group isomorphism; several examples.
Lagrange's Theorem.
Orthogonal operators on 3-dimensional real inner product
spaces.
Brief remarks about symmetry groups of geometric objects
in 3-space (not material for final exam).
Comments about permutation groups (not material for final exam).
Reading for week Feb 23-27: Section 7.3.
Reading for week March 2-6: Section 7.1.
Reading for week March 9-13: Section 7.1.
Reading for week March 16-20: Section 7.2.
Reading for week March 23-27:
Section 2.6 (dual spaces)
(Armstrong's book, or notes above) definition
of group, subgroup, normal subgroup, group homomorphism.
Reading for week March 30-April 3:
Basic properties of homomorphisms.
Orthogonal groups; rotations and reflections.
Dihedral groups.