Defenition and Propreties of Groups. Arithmetic Modulo n, permutations, cyclic groups. Cosets and Lagrange's theormem. Normal subgroups, homomorphisms, quotien groups, isomorphism theorems.
Propreties of the real numbers. Supremum and infimum, lim sup and lim inf. Sequences and convergence. The topology of Rn. Compactness, Heine-Borel. Continuity and Uniform Continuity. Derivatives, the Riemann integral, the fundamental theorem of calculus, improper integrals. Sequences and series of functions, uniform convergence. Fourier series.
Vector spaces, direct sums and complement of subspaces, linear maps, representation of linear maps by matrices, dual spaces, transpose mappings, multilinear mappings, determinants, inner products, orthogonal projections, the Gram-Schmidt algorithm. Eigenvalues and eigenvectors, diagonalization of symmetric matrices. Symmetric forms and Hermitian forms. Spectral Theorem. Isometries. The emphasis of this course is on proving all results.
Derivatives as linear maps, the chain rule. The Clairaut-Schwarz theorem. Taylor's theorem. Implicit function theorem and the invertible function theorem. Extrema, critical points. Lagrange multipliers. Double and triple integrals, Fubini's theorem, polar, spherical and cylindrical coordinates. Change of variables. Line integrals, Green's theorem. Parametric surfaces and surface integrals. Curl and Stokes's theorem, existence of potentials. Divergence and Gauss's theorem. Applications.
Propositional logic, proofs. Induction, invariants and state machines. Number theory. Modular arithmetic. Sums and asymptotics. Reccurences. Counting. Directed graphs, undirected graphs and connectivity. Ramsey numbers.
Probability axioms and their consequences. Conditional probability and independence. Random variables, distributions and densities, moments, sampling distributions. Weak law of large numbers, sums of independent random variables, moment generating functions, convergence concepts, the central limit theorem.
Introduction to the theory of statistical inference. Parametric point and interval estimation. Maximum likelihood estimation. Properties of estimators. Principles of hypothesis testing. Confidence intervals and tests for means and proportions (one and two sample). Introduction to linear models. Use of statistical software. ANOVA and Chi-square tests. (Formerly MAT2375)