Introduction to Algebra and Analysis

Overview

Elementary logic and set theory, number systems (including integers, rationals, reals and complex numbers), bounds, supremums and infimums, basic combinatorics, functions.
Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non-negative reals and convergence tests.
Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function. Properties of continuous and differentiable functions. L'Hopital's rule, Rolle's theorem, mean-value theorem.
Matrices and systems of simultaneous linear equations, vector spaces, linear dependence, basis, dimension.

Learning Objectives

It is intended that students shall, on successful completion of the module, be able:
• to understand and to apply the basic of mathematical language;
• use the language of sets and maps and understand the basic properties of sets (finiteness) and maps (injectivity, surjectivity, bijectivity);
• demonstrate knowledge of fundamental arithmetical and algebraic properties of the integers (divisibility, prime numbers, gcd, lcm) and of the rationals;
• Solve combinatorial counting problems in a systematic manner.
• Understand the fundamental properties of the real numbers (existence of irrational numbers, density of Q, decimal expansion, completeness of R).
• Understand the notions of a sequence of real numbers, including limits, convergence and divergence.
• Define convergence of infinite series.
• Investigate the convergence of infinite series using convergence tests.
• Define limits of functions and define continuous functions.
• Prove that a function is continuous or discontinuous.
• Prove and apply basic properties of continuous functions including the intermediate value theorem and the existence of a maximum and a minimum on a compact interval.
• Define a differentiable function and a derivative.
• Prove whether a function is differentiable.
• Calculate (using analysis techniques) derivatives of many types of functions.
• Understand, apply and prove Rolle's theorem and the Mean Value Theorem.
• Prove the rules of differentiation such as the product.
• Understand and apply the theory of systems of linear equations.
• Produce and understand the definitions of vector space, subspace, linear independence of vectors, bases of vector spaces, the dimension of a vector space.
• Apply facts about these notions in particular examples and problems.
• Understand the relation between systems of linear equations and matrices.

Skills

• Understanding of part of the main body of knowledge for mathematics: analysis and linear algebra.
• Logical reasoning.
• Understanding logical arguments: identifying the assumptions made and the conclusions drawn.
• Applying fundamental rules and abstract mathematical results, equation solving and calculations; problem solving.

Assessment

Computer Test