LA I: Epiphany Term Lecture Summaries
The table below lists the main topics to be covered in each lecture, and will be updated as the term proceeds.
(You may need to press shift-reload to get the latest version.)
| MATH1091 | Monday 11am CG93 | Tuesday 11am CG93 | Thursday 10am W103 |
| MATH1071 | Monday 12pm PH8 | Thursday 12pm PH8 | Friday 10am PH8 |
| 11 (26) | Collections exam |
1 Eigenvalues, eigenvectors and diagonalisation Changing the basis for the matrix of a linear transformation Eigenvectors and eigenvalues The characteristic equation |
Algebraic and geometric multiplicities of eigenvalues Eigenspaces |
| 12 (27) |
The Cayley-Hamilton theorem Equivalence relations and similarity of matrices Diagonalisable matrices |
A key result: an NxN matrix is diagonalisable ⇔ it has N linearly independent eigenvectors |
Examples; application to systems of ODEs |
| 13 (28) |
Linear independence of eigenvectors corresponding to distinct eigenvalues |
2 Inner product spaces Bilinear forms on real vector spaces Inner products on real vector spaces |
Problems class |
| 14 (29) |
The matrix of an inner product Symmetric and antisymmetric matrices Positive-definite matrices and Sylvester's criterion |
Complex (Hermitian) inner products Hermitian and anti-Hermitian matrices |
Norms on real vector spaces The triangle and Cauchy-Schwarz inequalities |
| 15 (30) |
The complex case and the complex C-S inequality Orthogonal and orthonormal vectors The Gram-Schmidt procedure |
The Gram-Schmidt procedure (ctd) | Problems class |
| 16 (31) |
The orthogonal complement of a subspace Orthogonal projection Properties of projection operators Projection as a way to find the element of a subspace closest to a given vector |
Bessel's inequality Orthogonal and unitary diagonalisation The orthogonal and special orthogonal groups The unitary and special unitary groups |
Symmetric and Hermitian matrices have: - real eigenvalues - orthog. eigenvectors for distinct eigenvalues Furthermore they can always be orthogonally or unitarily diagonalised |
| 17 (32) |
Example: diagonalisation of a symmetric matrix 3 Orthogonal polynomials Linear ODEs with polynomial solutions |
More-general (weighted) inner products on spaces of functions Legendre polynomials as an orthonormal system Further examples: Chebyshev I & II, Hermite and Laguerre |
Problems class |
| 18 (33) |
Symmetric linear operators Example: the Laguerre differential operator Linear ODEs as eigenvalue problems |
Orthogonal polynomials w.r.t. an inner product (,), graded by degree, are eigenfunctions of certain linear differential operators which are symmetric w.r.t. the same inner product Polynomial solutions of the Legendre equation |
Proof that orthogonal polynomials are eigenfunctions of certain symmetric linear differential operators Hermitian and anti-Hermitian operators on complex inner product spaces |
| 19 (34) |
4 Group theory The group axioms: closure, associativity, identity and inverses Examples of groups Subgroups |
More examples Isomorphism of groups Finite groups and Lagrange's theorem The cyclic group ℤn |
Problems class |
| 20 (35) |
Group (Cayley) tables The Klein four-group Direct products Rotations as (special) orthogonal transformations |
The dihedral group The symmetric and alternating groups Multiplicative groups |
SO(3) and SU(2) Euler's principal axis theorem π1(SO(3)) = ℤ2 (not examinable!) |
 |
 |
 |
|
 |
 |
 |
|---|---|---|---|---|---|---|
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
|---|