# 1st Year

#### Course programme

In each of our courses (MORSE, MathStat, Data Science) students will be required to study core modules and can select optional modules. Course curriculum guidelines and core and optional module lists can be found in our first year course programme document. Descriptions of the core modules and most common optional modules are listed below.

###### Vectors & Matrices

Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector space, linear mapping and its representation by a matrix. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.

###### Calculus 1/2
Calculus is the mathematical study of continuous change. In this module there will be considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. By the end of the year you will be able to answer interesting questions like, what do we mean by `infinity’? This module is focused on developing your skills with calculations involving calculus.
###### Sets and Numbers

It is in its proofs that the strength and richness of mathematics is to be found. University mathematics introduces progressively more abstract ideas and structures, and demands more in the way of proof, until most of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.

###### Linear Algebra

Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector space, linear mapping and its representation by a matrix. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.

###### Mathematical Analysis 1/2
Analysis is the rigorous study of calculus. In this module, there will be a considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers, and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. The module will allow you to deal carefully with limits and infinite summations, approximations to pi and e, and the Taylor series. The module also covers construction of the integral and the Fundamental Theorem of Calculus.
###### Probability 1

Probability is a foundational module that will introduce you both to the important concepts in probability but also the key notions of mathematical formalism and problem-solving. Want to think like a mathematician? This module is for you. You will learn how to to express mathematical concepts clearly and precisely and how to construct rigorous mathematical arguments through examples from probability, enhancing your mathematical and logical reasoning skills. You will also develop your ability to calculate using probabilities and expectations by experimenting with random outcomes through the notion of events and their probability. You’ll learn counting methods (inclusion–exclusion formula and binomial co-efficients), and study theoretical topics including conditional probability and Bayes’ Theorem.

###### Probability 2

This module continues from Probability 1, which prepares you to investigate probability theory in further detail here. Now you will look at examples of both discrete and continuous probability spaces. You’ll scrutinise important families of distributions and the distribution of random variables, and the light this shines on the properties of expectation. You’ll examine mean, variance and co-variance of distribution, through Chebyshev's and Cauchy-Schwarz inequalities, as well as the concept of conditional expectation. The module provides important grounding for later study in advanced probability, statistical modelling, and other areas of potential specialisation such as mathematical finance.

###### Introduction to Statistical Modelling

This module is an introduction to statistical thinking and inference. You’ll learn how the concepts you met from Probability can be used to construct a statistical model – a coherent explanation for data. You’ll be able to propose appropriate models for some simple datasets, and along the way you’ll discover how a function called the likelihood plays a key role in the foundations of statistical inference. You will also be introduced to the fundamental ideas of regression. Using the R software package you’ll become familiar with the statistical analysis pipeline: exploratory data analysis, formulating a model, assessing its fit, and visualising and communicating results. The module also prepares you for a more in-depth look at Mathematical Statistics in Year Two.

###### Introduction to Quantitative Economics

The focus of this module is mainly on economic theory but "real world" applications of relevant theories will also be examined, subject to time limitations. The module covers aspects of microeconomics and macroeconomics. Microeconomics is concerned with the economic behaviour of individual consumers and producing firms, and their interaction in markets for goods, services and factors of production. Macroeconomics, on the other hand, is concerned with aggregate economic variables or the workings of the national economy as a whole such as Gross Domestic Product, unemployment, inflation and interest rates, and with government economic policies to influence these variables.

###### Programming for Computer Scientists

On this module, whatever your starting point, you will begin your professional understanding of computer programming through problem-solving, and fundamental structured and object-oriented programming. You will learn the Java programming language, through practical work centred on the Warwick Robot Maze environment, which will take you from specification to implementation and testing. Through practical work in object-oriented concepts such as classes, encapsulation, arrays and inheritance, you will end the course knowing how to write programs in Java, and, through your ability to analyse errors and testing procedures, be able to produce well-designed and well-encapsulated and abstracted code.

###### Design of Information Structures

Following on from Programming for Computer Scientists, on the fundamentals of programming, this module will teach you all about data structures and how to program them. We will look at how we can represent data structures efficiently and how we can apply formal reasoning to them. You will also study algorithms that use data structures. Successful completion will see you able to understand the structures and concepts underpinning object-oriented programming, and able to write programs that operate on large data sets.

###### Mathematical Programming I

Operational Research is concerned with advanced analytical methods to support decision making, for example for resource allocation, routing or scheduling. A common problem in decision making is finding an optimal solution subject to certain constraints. Mathematical Programming I introduces you to theoretical and practical aspects of linear programming, a mathematical approach to such optimisation problems.

##### Optional modules

List of optional module names for each of our courses and examples for unusual options can be found in the first year course programme document.