group homomorphisms and isomorphism theorems definition and examples of groups and their properties This module will introduce you to some of the main concepts and techniques of the subject. A pplicat ions of group theory can be found in almost every area of mathematics, and also in chemistry and physics. Groups usually arise in connection with invariance properties of the objects under study, e.g., the collection of all geometrical transformations of the Euclidean plane that leave invariant certain figure on it, form a group. It is quite different from Linear Algebra and i ts origins can be traced back to the work of Lagrange on permutation groups. Group Theory is the branch of Algebra concerned with the study of groups. Pre-requisite: MTH1011 Introduction to Algebra and Analysis, MTH2011 Linear Algebra continuous mappings between metric spaces open sets, closed sets, closure points, sequential convergence, density, separability definition and examples of metric spaces (including function spaces) This notion allows us to define convergence and continuity in a much more abstract setting, and induces topological properties, like open sets and closed sets which lead to the study of more abstract topological spaces. The most familiar example is the real line, with the distance from x to y given by |x- y|. A metric space is a set with a notion of distance, called a metric. ![]() The aim of this module is to move gradually away from real numbers to the more general setting of metric spaces. Pre-requisite: MTH2011 Linear Algebra, MTH2012 AnalysisĪnalysis in Semester 1 is the study of convergence and continuity, it is fundamentally linked to the structure of the real numbers. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus. Mean value theorems including that of Cauchy, proof of l'Hôpital's rule, Taylor's theorem with remainder. Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted). Its primary aim is to equip you with some key techniques of mathematical analysis, so that you can apply them both within mathematics itself and beyond in various other disciplines.Ĭauchy sequences, especially their characterisation of convergence. ![]() This module extends and develops several core ideas from Level 1 analysis and seeks to deepen your understanding of them. Pre-requisite: MTH1011 Introduction to Algebra and Analysis and MTH1021 Mathematical Methods 1 Basic computer application of linear algebra techniques.Īdditional topics and applications, such as: Schur decomposition, orthogonal direct sums and geometry of orthogonal complements, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form. Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties. Inner product spaces, orthogonality, Cauchy-Schwarz inequality. Change of basis, diagonalization, similarity transformations. Matrix inversion, definition and computation of determinants, relation to area/volume. Matrix representation of linear maps, eigenvalues and eigenvectors of matrices. ![]() ![]() Linear transformations, image, kernel and dimension formula. Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension. The techniques of linear algebra are core to most areas of mathematics, statistics, data analysis, physics and computer science. This module will build upon the level 1 linear algebra to develop the theory, methods and algorithms needed throughout the degree programme and beyond. Pre-requisites: MTH1011 Introduction to Analysis and Algebra and MTH1021 Mathematical Methods 1
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