Michaelmas
Lecture 1
Introductory remarks. Definitions and notation, Elementary set theory: subsets, the null set; union, intersection, difference and complement of sets, an easy proposition based on these definitions; Cartesian product
Lecture 2
Relations: definition and properties (reflexivity, symmetry, antisymmetry, transitivity); partial order and linear order relations; Hausdorff maximality priniciple; equivalence relations and examples.
Lecture 3
Mappings: definition, notation, examples; binary operations, projections;
Lecture 4
Definition of one-to-one (injective) and onto (surjective) maps; bijections; examples and counterexamples of bijections. Composition.
Lecture 5
Composition of maps, an associative law for composition. Further properties relating to composition of one-to-one, onto and bijective maps. Definition of the set of all one-to-one mappings.
Lecture 6
Maps and functions. Examples. Introduction to the set of integers. Associativity, commutativity, existence of an identity and inverse under addition. Associativity, commutativity, existence of an identity under multiplication. Least integer principle.
Lecture 7
Uniqueness of least integer. The principle of induction. The Euclidean Algorithm. The greatest common divisor: definition, uniqueness and existence.
Lecture 8
Lecture 9
Relative primes, prime numbers, no largest prime and other results. Unique factorisation theorem. Discussion of modern applications in number theory and encryption.
Lecture 10
Congruence modulo an integer, congruence classes, Z_n. Beginning group theory.
Lecture 11
The group axioms, abelian groups. Examples: S_3, cyclic groups, Z_n, GL(n).
Lecture 12
Show some properties of groups: unique identity element, unique inverse for each element etc.
Lectures 18&19
Permutations: as a composition of disjoint cycles; as a composition of transpositions. Definition of odd & even permutations.
Congruences and left cosets: Proved that left cosets have the same number of elements and that they partition the group. Lagrange's theorem and the fact that the order of an element divides the order of the group.
Definition of the order of an element in terms of the cyclic subgroup the element generates. I also gave some examples using G=S_3 and H= cyclic subroup of order 3 (finding the left cosets, the order of some elements etc). Decomposing a partition into transpositions to determine parity.
Lectures 20
Permutation matrix: definition and examples. More on cosets (left and right).
Lectures 21
Properties of cosets. Lagrange's theorem (again) and corollaries: including the result that groups of prime order are cyclic.
Lectures 22 & 23
Normal subgroups. Definition in terms of cosets and as gNg^{-1} and examples. The product of cosets and the quotient group defined from the set of cosets. The group axioms for the quotient group and an example based on the group D_8 of symmetries of the square. Beginning discussion of isomorphisms.
Lectures 24
Example of isomorphic groups, from their Cayley tables. Definition and examples of group homomorphisms.
Lectures 25
Definition and examples of the kernel of a homomorphism.
Lectures 26 & 27
Proved the kernel of a homomorphism from G to R is a normal subgroup of G. Stated the fundamental homomorphism theorem and started the proof.

Lectures 29
A little more on the fundamental homomorphism theorem including an example with Z_6 and Z_3. Definition of a ring.
Lectures 30 & 31
Examples of rings: Commutative and noncommutative (quaternions). Leading to the definition of a division ring and a field. Proved some easy properties of rings.
Hilary Lecture 1
Linear Independence: definition and examples and some additional properties of linearly independent vector spaces. Definition of a basis set of vectors and examples.
Lecture 2
Dimension of a vector space and subspaces. Mappings between vector spaces.
Lecture 3
Changing bases and coordinate systems ...
Lecture 4

week 2 - No Lectures

Lecture 5
Cauchy-Schwarz inequality, orthgonality and inner products, orthogonal complements, orthonormal basis set.
Lecture 6
The cross product
Lecture 7 and 8
Linear transformations, definition and examples. The induced matrix: matrix-vector multiplication. Definitions of invertiblity, singularity and characteristic roots.
Lecture 9
Linear transformations continued: invertible and singular. Characteristic root (eigenvalue). For matrices: a law for multiplying a column vector by a matrix.
Lecture 10
Matrix Algebra and solving linear systems, forming the augmented matrix. Solving with elementary row operations (defined). Properties of row operations (invertibility etc).
Lecture 11 and 12
Solving systems of linear equations. Definition of row echelon form and reduced row echelonform. Examples of Gaussian elimination and Gauss-Jordan elimation. Comment on homogeneous linear equations. Definition of rank.
Lecture 13
Definitions of transpose (and some properties) and trace. Definition of row equivalence and elementary matrices.
Lecture 14
Recap of Gauss-Jordan strategy to solve a system of linear equations. Discussion and examples of the link between solutions and linear independence of rows and colums of the matrix, A.
Lecture 15 and 16
Theorem on equivalent statements to "A is invertible". Special matrices and their inverses: diagonal, triangular (upper and lower). Proofs of some properties of these matrices eg. on the invertibility of diagonal and triangular matrices, products and inverses of upper and lower triangular matrices.
Lecture 17
Special matrices continued: symmetric matrices.
Lecture 18
Matrix factorisation ie LU decomposition: method and examples. Using LU decompostion to solve systems of linear equations. Crout's algorithm.
Lecture 19 and 20
Definition of row and column spaces. Theorem: Rank(A)=dim R(A) = dim C(A). Finding bases for the row and column spaces of a matrix. Nullspaces.
Lecture 21
Determinants: definition and properties (presented as theorems). The effects of ero's on determinants.
Lecture 22
Det(A) = Det(A transpose). Cramer's rule. The determinant and criteria for invertibility
Lecture 23 and 24
Eigenvalues and eigenvectors. Definitions and examples. The characteristic equation. Determinant and trace in terms of eigenvalues. Linear independence of the eigenvectors corresponding to difference eigenvalues of a matrix. Decompostion (diagonalisation) of A in terms of matrices of eigenvalues and eigenvectors. Using this to calculate A raised to a power.
Lecture 25
Diagonalisation - proof by construction. Link with linear-independence of eigenvectors.
Lecture 26
Applications: finding solutions to coupled differential equations. Remark on complex eigenvalues. Proof that symmetric matrices have only real eigenvalues.
Lecture 27 and 28
Orthogonal matrices and diagonalisation. Definition of orthogonal matrices and examples including rotation matrices. Recap of orthonormal bases and proof relation orthogonality and orthonormal basis of row vectors. Diagonalisation of orthogonal matrices (including Gram-Schmidt method). Examples.
Lecture 29
Applications of Orthogonal matrices: QR decomposition, method and examples.
Lecture 30
Least squares method to solve an inconsistent system of equations. Motivation by looking at fitting a line to a set of points. Method including a definition of the associated normal system. Example. Using QR decomposition and least squares.
Trinity Lecture 1
Cancelled - school review
Lecture 2
Band and block matrices
Lecture 3 and 4
Good Friday
Lecture 5
Tutorial
Lecture 6
Introduction to jordan normal form
Lecture 7 and 8
Details of the transformation of a matrix to its jordan form. 3 worked examples in increasing complexity.
Lecture 9
Further examples of Jordan form. Discussion and illustration of the sensitivity of Jordan form to perturbations. Applications of Jordan form: (i) calculating matrix inverses, (ii) determining powers of a matrix in Jordan form, (iii) exponentiation of a matrix in Jordan form.
Lecture 10
Matrix norms, common examples. Condition number: definition and interpretation.
Lecture 11 and 12
Cayley-Hamilton Theorem. Examples and a proof (using Jordan form).

Lecture 13
(ii) Cayley-Hamilton to determine analytic functions of a matrix, with example. (iii) Computing exp(At), with example.
Lecture 14 and 15
Discussion of method for (iii) if 1 or more eigenvalues are repeated. Hermition Matrices: definition and example. Discussion of notation and recap of complex conjugation. Properties of hermitian matrices eg all real diagonal elements. Examples including Pauli matrices. Eigenvalues of a Hermitian matrix. Definition of anti-hermitian and unitary matrices.
Note that this material will not be examined.

Lecture 16
(Real) Quadratic Forms. Definition and example. Definition of positive definite and positive semi-definite. Properties of quadratic forms including (i) diagonalisation and (ii) positive definiteness.
Lecture 17
Application: statistical analysis of multivariate data ie. how to maximise a quadratic form given a constraint. Method and example.
Lecture 18 and 19
Summary of quadratic forms. Discussion of conic sections as applications of quadratic forms. Sylvester's inertia theorem.