# PrEMAS 1 - Mathematical methods for actuaries

## In this course, we will explore the basic concepts of linear algebra such as vector-and matrix operations and the mathematical analysis on functions of multiple variables, like differentiation and integration in higher dimensions.

Prior knowledge of single variable calculus and elementary geometric theorems is recommended. The course will provide an introduction in the theory of linear differential equations. Concepts like (power)series and complex numbers are necessary to understand this subject. The theory of linear differential equations and the operations on multivariable functions has many applications, especially useful for the actuary.

### Entry Requirments

General enty requirements for the premaster.

### Learning objectives

Learning objectives are derived from the set of competences, adopted by the Koninklijk Actuarieel Genootschap (AG). The course contains of two parts: Linear Algebra and Advanced Analytics.

##### Linear Algebra

Matrices and systems of linear equations

With regard to matrices and systems of linear equations, one is able to:

• Deal with elementary operations on matrices (adding, scalar multiplication, composing, transposing);
• Calculate the determinant of a matrix and apply Cramers rule to solve a system of linear equations;
• Use the Gaussian elimination method to determine the rank of a matrix, to compute the inverse of a matrix and to solve a system of linear equations;
• Determine the characteristic polynomial, eigenvalues and eigenvectors of a matrix;
• Determine whether if a given matrix is diagonalizable and, if so, find a diagonalizing matrix;

Vectors, vectorspaces and innerproduct spaces

With regard to vectors, vectorspaces, and innerproduct spaces, one is able to:

• Deal with elementary operations on vectors (adding, scalar multiplication, multiplication (inner and outer);
• Explain the concepts of a vectorspace, innerproduct space and orthogonality.

Parametric representations

In terms of parametric representations and functions, you are able to:

• Describe the difference between a function and a curve;
• Draw a parametric representation in polar coordinates;
• Give examples of curves in polar coordinates;
• Determining the horizontal and vertical tangents of a parametric representation in polar coordinates;
• Determining where the curve intersects the axes of a polar parametric representation;
• Determine points of intersection between curves in polar coordinates;
• Analyze any asymptotic behavior of a parametric representation in polar coordinates;
• Calculate areas and lengths of curves in polar coordinates;
• Give the mathematical definition of the concepts convex and concave;
• Determine on the basis of criteria whether a curve has a convex or concave course;
• Define continuity in your own words (continuity of functions from IR2 to IR);
• Determine whether a function in multiple variables is continuous in a point;
• Demonstrate discontinuity of a function in a point by means of a counterexample (proof of continuity).

Series

In terms of series, you are able to:

• Explain the difference between a sequence and a series;
• Solve a recursive sequence with given starting values;
• Examples give the following types of series: geometric series, alternating series, power series;
• Define the geometric and harmonic series and determine the sum of these series;
• Write down the formula of the 'p-series' and the associated convergence properties;
• Be able to describe the various convergence tests;
• Indicate for series and their sum whether there is (relative) convergence or divergence by means of convergence tests (divergence test, minorance, majorance, test for alternating series, ratio test, square root test, integral test);
• Describe the concept of power series;
• Determining power series on the basis of various tests of convergence interval and radius of convergence (divergence test, test for alternating series, ratio test, square root test, integral test);
• Approximate series numerically and thereby determine a margin of error;
• Define a Taylor and MacLaurin series, and know how the residual term is determined;
• Represent a function as a Taylor series and calculate its convergence interval and radius.

Differentiation

In terms of differentiation, you are able to:

• Determine the first and second order partial derivatives of a multivariate function, both directly and implicitly;
• Determining the partial derivative of a function using the limit definition of a partial derivative;
• Analyzing the zeros and the stationary points of a multivariate function;
• Being able to determine linearizations (planes) of functions from IR2 to IR;
• From a function of IR2 to IR determine the Jacobian and the Hessian;
• Determine on the basis of the determinant whether a stationary point is a saddle point, a (local or absolute) minimum or maximum.
• Apply chain rule(s) for a multivariate function;
• Solve simple optimization problems using the theory of gradients and Lagrange multipliers (Euler-Lagrange method).

Integration

In terms of integration, you are able to:

• Determine and solve the integral formulas of areas, volumes and lengths;
• Distinguish normal areas and thus solve integrals;
• Switching to different coordinate systems, especially between spherical, cylinder and polar coordinates or linear transformations;
• When solving an integral, use alternative coordinate systems, making use of the Jacobian;
• Write down the definition of Fubini's theorem and apply this theorem to solving a multiple integral.

Complex numbers

In the field of complex numbers, you are able to:

• Describe the concept of a complex number, and express a complex number of the form a + b.i into the polar form z = r.(cos θ + i.sin θ);
• To be able to describe the concepts of modulus and argument and to be able to apply them to complex numbers;
• Describe and apply De Moivre's theorem to prove simple theorems such as the double angle formula for sin(2x);
• Be able to describe a complex number in the exponential form;
• Perform simple operations (addition, subtraction, multiplication, division, exponentiation) of complex numbers.

Differential equations

In the field of differential equations, you are able to:

• Describe what a (first order) differential equation is;
• Give examples of some applications of (first order) differential equations;
• Finding and approaching solutions by separating equations and Euler's Method;
• Determining solutions through the method of Frobenius: power series developments;
• Indicate the difference between first and second order linear differential equations and between homogeneous and inhomogeneous linear differential equations;
• Solve second order inhomogeneous linear differential equations with constant terms, also with complex solutions.

### Literature

Mandatory literature to be purchased by student:

James Stewart, Roland Minton Calculus, Early Transcendentals Metric Edition 9th Edition, March 2020, ISBN 9780357113516

Linear algebra

Gilbert Strang, Linear Algebra and its Applications – (fourth edition) – Academic Press Inc – ISBN 978-0-03-010567-8

Literature made available by the Actuarial Institute:

• Trigonometric Identities Sheet
• Sheet with standard derivatives
• Sheet with step-by-step plan for testing criteria series
• Sheet with standard Taylor and MacLaurin power series developments
• Sheet with standard primitives
• Standard limits sheet

### Teachers

• Marcel Roggeband - Linear algebra
• Sjoert Fleurke - Advanced Analytics

### Important to know

• Lectures start on Monday 28 August 2023 from 14.30-17.30 hours. The first six lectures are devoted to Linear Algebra and the remaining 11 lectures to Advanced Analytics. On Thursday 14 September 2023 there will be an extra lecture for Linear Algebra. The complete schedule will be published in the group page.
• Exams are planned on:
- Linear Algebra: 2 October 2023
- Resit Linear Algebra: 30 October 2023
- Advanced Analytics: 18 December 2023
- Resit Advanced Analytics: 5 February 2024
Changes reserved
• We advise you to practice the assignments in the study guide. The assignments are discussed in the lecture.
• Available literature from the Actuarial Institute will be publised in the group page. Access to the group page will be given after enrollment.
• Students have passed the module when the result for both parts is at least 5.5.
• The costs for this module are € 3.525.