Prelims Mathematics & Philosophy 2019-20

Foreword:

Synopses

The synopses give some additional detail, and show how the material is split between the different lecture courses. They also include details of recommended reading.

Syllabus:

The syllabus here is that referred to in the Examination Regulations 2019 Special Regulations for the Preliminary Examination in Mathematics & Philosophy (https://www.admin.ox.ac.uk/examregs/).

Examination Conventions can be found at: http://www.maths.ox.ac.uk/members/students/undergraduate-courses/examina....

Mathematics I

Sets: examples including the natural numbers, the integers, the rational numbers, the real numbers; inclusion, union, intersection, power set, ordered pairs and cartesian product of sets. Relations. Definition of an equivalence relation.

The well-ordering property of the natural numbers. Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.

Maps: composition, restriction, injective (one-to-one), surjective (onto) and invertible maps, images and preimages.

Systems of linear equations. Matrices and the beginnings of matrix algebra. Use of matrices to describe systems of linear equations.
Elementary Row Operations (EROs) on matrices. Reduction of matrices to echelon form. Application to the solution of systems of linear equations.

Inverse of a square matrix. Reduced row echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method. Transpose of a matrix; orthogonal matrices.

Vector spaces: definition of a vector space over a field (such as $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}$). Subspaces. Many explicit examples of vector spaces and subspaces.

Span of a set of vectors. Examples such as row space and column space of a matrix. Linear dependence and independence. Bases of vector spaces; examples. The Steinitz Exchange Lemma; dimension. Application to matrices: row space and column space, row rank and column rank. Coordinates associated with a basis of a vector space.

Use of EROs to find bases of subspaces. Sums and intersections of subspaces; the dimension formula. Direct sums of subspaces.

Linear transformations: definition and examples (including projections associated with direct-sum decompositions). Some algebra of linear transformations; inverses. Kernel and image, Rank-Nullity Theorem. Applications including algebraic characterisation of projections (as idempotent linear transformations).

Matrix of a linear transformation with respect to bases. Change of Bases Theorem. Applications including proof that row rank and column rank of a matrix are equal.

Bilinear forms; real inner product spaces; examples. Mention of complex inner product spaces. Cauchy--Schwarz inequality. Distance and angle. The importance of orthogonal matrices.

Introduction to determinant of a square matrix: existence and uniqueness and relation to volume. Proof of existence by induction. Basic properties, computation by row operations.

Determinants and linear transformations: multiplicativity of the determinant, definition of the determinant of a linear transformation. Invertibility and the determinant. Permutation matrices and explicit formula for the determinant deduced from properties of determinant.

Eigenvectors and eigenvalues, the characteristic polynomial. Trace. Proof that eigenspaces form a direct sum. Examples. Discussion of diagonalisation. Geometric and algebraic multiplicity of eigenvalues.

Gram-Schmidt procedure.

Spectral theorem for real symmetric matrices. Matrix realisation of bilinear maps given a basis and application to orthogonal transformation of quadrics into normal form. Statement of classification of orthogonal transformations.

Axioms for a group and for an Abelian group. Examples including geometric symmetry groups, matrix groups ($GL_{n}$, $SL_{n}$, $O_{n}$, $SO_{n}$, $U_{n}$), cyclic groups. Products of groups.

Permutations of a finite set under composition. Cycles and cycle notation. Order. Transpositions; every permutation may be expressed as a product of transpositions. The parity of a permutation is well-defined via determinants. Conjugacy in permutation groups.

Subgroups; examples. Intersections. The subgroup generated by a subset of a group. A subgroup of a cyclic group is cyclic. Connection with hcf and lcm. Bezout's Lemma.

Recap on equivalence relations including congruence mod n and conjugacy in a group. Proof that equivalence classes partition a set. Cosets and Lagrange's Theorem; examples. The order of an element. Fermat's Little Theorem.

Isomorphisms, examples. Groups of order 8 or less up to isomorphism (stated without proof). Homomorphisms of groups with motivating examples. Kernels. Images. Normal subgroups. Quotient groups; examples. First Isomorphism Theorem. Simple examples determining all homomorphisms between groups.

Group actions; examples. Definition of orbits and stabilizers. Transitivity. Orbits partition the set. Stabilizers are subgroups.

Orbit-stabilizer Theorem. Examples and applications including Cauchy's Theorem and to conjugacy classes.

Orbit-counting formula. Examples.

The representation $G\rightarrow \mathrm{Sym}(S)$ associated with an action of $G$ on $S$. Cayley's Theorem. Symmetry groups of the tetrahedron and cube.

Mathematics II

Real numbers: arithmetic, ordering, suprema, infima; real numbers as a complete ordered field. Countable sets. The rational numbers are countable. The real numbers are uncountable.

The complex number system. The Argand diagram; modulus and argument. De Moivre's Theorem, polar form, the triangle inequality. Statement of the Fundamental Theorem of Algebra. Roots of unity. Simple transformations in the complex plane. Polar form, with applications.

Sequences of (real or complex) numbers. Limits of sequences of numbers; the algebra of limits. Order notation.

Subsequences; every subsequence of a convergent sequence converges to the same limit. Bounded monotone sequences converge. Bolzano-Weierstrass Theorem. Cauchy's convergence criterion. Limit point of a subset of the line or plane.

Series of (real or complex) numbers. Convergence of series. Simple examples to include geometric progressions and power series. Alternating series test, absolute convergence, comparison test, ratio test, integral test.

Power series, radius of convergence, important examples to include definitions of relationships between exponential, trigonometric functions and hyperbolic functions.

Continuous functions of a single real or complex variable. Definition of continuity of real valued functions of several variables.

The algebra of continuous functions. A continuous real-valued function on a closed bounded interval is bounded, achieves its bounds and is uniformly continuous. Intermediate Value Theorem. Inverse Function Theorem for continuous strictly monotonic functions.

Sequences and series of functions. The uniform limit of a sequence of continuous functions is continuous. Weierstrass's M-test. Continuity of functions defined by power series.

Definition of derivative of a function of a single real variable. The algebra of differentiable functions. Rolle's Theorem. Mean Value Theorem. Cauchy's (Generalized) Mean Value Theorem. L'Hôpital's Formula. Taylor's expansion with remainder in Lagrange's form. Binomial theorem with arbitrary index.

Step functions and their integrals. The integral of a continuous function on a closed bounded interval. Properties of the integral including linearity and the interchange of integral and limit for a uniform limit of continuous functions on a bounded interval. The Mean Value Theorem for Integrals. The Fundamental Theorem of Calculus; integration by parts and substitution.

Term-by-term differentiation of a (real) power series (interchanging limit and derivative for a series of functions where the derivatives converge uniformly).

Mathematics III(P)

General linear homogeneous ODEs: integrating factor for first order linear ODEs, second solution when one solution is known for second order linear ODEs. First and second order linear ODEs with constant coefficients. General solution of linear inhomogeneous ODE as particular solution plus solution of homogeneous equation. Simple examples of finding particular integrals by guesswork.

Partial derivatives. Second order derivatives and statement of condition for equality of mixed partial derivatives. Chain rule, change of variable, including planar polar coordinates. Solving some simple partial differential equations (e.g. $f_{xy} = 0$, $f_x = f_y$).

Parametric representation of curves, tangents. Arc length. Line integrals.

Jacobians with examples including plane polar coordinates. Some simple double integrals calculating area and also $\int_{\mathbb{R}^2} e^{-(x^2+y^2)} dA$.

Simple examples of surfaces, especially as level sets. Gradient vector; normal to surface; directional derivative; $\int^B_A \nabla \phi \cdot d\mathbf{r} = \phi(B)-\phi(A)$.

Taylor's Theorem for a function of two variables (statement only). Critical points and classification using directional derivatives and Taylor's theorem. Informal (geometrical) treatment of Lagrange multipliers.

Sample space, algebra of events, probability measure. Permutations and combinations, sampling with or without replacement. Conditional probability, partitions of the sample space, theorem of total probability, Bayes' Theorem. Independence.

Discrete random variables, probability mass functions, examples: Bernoulli, binomial, Poisson, geometric. Expectation: mean and variance. Joint distributions of several discrete random variables. Marginal and conditional distributions. Independence. Conditional expectation, theorem of total probability for expectations. Expectations of functions of more than one discrete random variable, covariance, variance of a sum of dependent discrete random variables.

Solution of first and second order linear difference equations. Random walks (finite state space only).

Probability generating functions, use in calculating expectations. Random sample, sums of independent random variables, random sums. Chebyshev's inequality, Weak Law of Large Numbers.

Continuous random variables, cumulative distribution functions, probability density functions, examples: uniform, exponential, gamma, normal. Expectation: mean and variance. Functions of a single continuous random variable. Joint probability density functions of several continuous random variables (rectangular regions only). Marginal distributions. Independence. Expectations of functions of jointly continuous random variables, covariance, variance of a sum of dependent jointly continuous random variables.