The foundational mathematical structures required for further study will be introduced. This includes an introduction to set theory, functions, inequalities, vectors, matrices, complex numbers, polynomials equations, and elementary properties of the integers. Some proofs will be covered to demonstrate how to construct different styles of proof. A working knowledge of calculus in terms of differentiation and integration of functions of a single variable will be developed. Linear algebra will form a large part of the module starting from matrices and systems of linear equation, then moving on the vector spaces and linear transformations.
Learning Outcomes
By the end of the module students should be able to:
Appreciate different methods of proof used in mathematics, including proof by induction and proof by contradiction, and be able to construct some proofs.
Understand the basic theory of set and functions.
Solve basic inequalities, including those involving quadratic terms and moduli.
Perform vector calculations, including scalar and vector products, and describe lines and planes in terms of vectors.
Work with complex numbers and perform standard calculations.
Understand and work with basic theory of polynomials and of integers.
Calculate derivatives and integrals of functions of a real variable using standard techniques, and apply differentiation and integration in appropriate situations.
Perform matrix calculations including reducing to echelon form, calculating inverses and determinants, and use matrix methods to solve systems of linear equations.
Make calculations with permutations including determining the sign of a permutations.
Determine eigenvalues and eigenvectors and diagonalise matrices where possible.
Understand and use the basic concepts of linear algebra (over the real numbers) and be able to construct elementary proofs in the development of the theory.
Understand the theory of linear transformations, determinants, eigenvectors, characteristic polynomials, minimal polynomials and be able to make calculations in examples.
Understand the basic theory of inner products and apply it to questions of orthogonality and diagonalizability of self-adjoint transformations.
Assessment
Assessment Methods & Exceptions
Assessment:
2hr examination (40%) In--course assessment (60%) (including a variety of assessment possibly including problem sheets, class tests, in-tutorial assessments, online quizzes, and group projects)
