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Prof Adam Ostoja-Ostaszewski and Dr James Ward

This course is compulsory on the BSc in Actuarial Science, BSc in Business Mathematics and Statistics, BSc in Financial Mathematics and Statistics, BSc in Mathematics and Economics, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available on the BSc in Economics, BSc in Management and MSc in Economics (2 Year Programme). This course is available as an outside option to students on other programmes where regulations permit. This course is available with permission to General Course students.

Students should ideally have taken the course Mathematical Methods (MA100) or equivalent, entailing intermediate-level knowledge of calculus (proficiency in techniques of differentiation and integration) and linear algebra (including linear independence, eigenvalues and diagonalisation).

This course develops ideas first presented in MA100. It is divided into two halves: calculus and linear algebra. The calculus half explores how integrals may be calculated or transformed by a variety of manipulations, and how they may be applied to the solution of differential equations. This aim is achieved by studying the following topics: Limit calculations. Riemann integral. Multiple integration. Improper integrals. Manipulation of integrals. Laplace transforms. Riemann-Stieltjes integral, to a level of detail dependent on time constraints. The linear algebra half covers the following topics: Vector spaces and dimension. Linear transformations, kernel and image. Real inner products. Orthogonal matrices, and the transformations they represent. Complex matrices, diagonalisation, special types of matrix and their properties. Jordan normal form, with applications to the solutions of differential and difference equations. Singular values, and the singular values decomposition. Direct sums, orthogonal projections, least square approximations, Fourier series. Right and left inverses and generalized inverses.

This course is delivered through a combination of classes and lectures totalling a minimum of 60 hours across Michaelmas and Lent Terms.

Written answers to set problems will be expected on a weekly basis.

Useful background texts:

(i) for the calculus half:

Adam Ostaszewski, Advanced Mathematical Methods, (Cambridge University Press 1999)

Ken Binmore and Joan Davies, Calculus, Concepts and Methods (Cambridge University Press 2002);

Robert C. Wrede and Murray R. Spiegel, Advanced Calculus (McGraw-Hill Education; 3rd edition 2010).

(ii) for the linear algebra half:

Martin Anthony and Michele Harvey, Linear Algebra: Concepts and Methods (Cambridge University Press 2012).

Exam (100%, duration: 3 hours) in the summer exam period.