This course covers fundamental concepts in pure mathematics. Building on the definition of group given in level 1, deeper properties of these objects will be studied. Student's intuition for groups will developed through examples. The abstract concept of vector spaces, and linear transformations between these spaces will be introduced. By considering basis for vector spaces, the concept of linear transformation will be related to that of matrices. In the final part the course, groups and linear transformations come together. This is done by considering the symmetries of spaces and shapes.

2 x lectures per week. Fortnightly tutorials.

One degree examination (80%) (1 hour 30 mins); coursework (20%).

Reassessment will not be available for continuous assessment.  Immediate feedback will be given to students at the assignment deadline, and it is not practical to set multiple versions of each of these exercises.

In total 80% of the assessment can be repeated for students not reaching the threshold grade.

Note that students are required to submit 75% of the assessments in order to receive a grade for the course. Accordingly, students who have a substantial period of illness and have all feedback exercises and tutorial group work MV'ed, will not be in danger of failing to receive credits

The course covers fundamental concepts in pure mathematics. The aim of the first part of the course is to develop the student's understanding of the concept of group. Throughout, a strong emphasis is put on examples in order to develop students' intuition for these objects. The aim of the second part of the course is to introduce abstract vector spaces and understand their basic properties. Linear transformations between abstract vector spaces will be defined and the role of basis in relating linear transformations to matrices will be explained. The final aim of the course is to tie together the concept of group and linear transformation by studying the action of groups on geometric objects. This is done by considering the symmetries of spaces and shapes.

By the end of this course, students will be able to:

■ Determine whether a given function is injective, surjective or bijective, and produce examples of each type of function.

■ State the definition of a group and a subgroup; produce examples of abelian and non-abelian groups; state, prove and apply Lagrange's theorem.

■ Prove that conjugacy is an equivalence relation in any group, and apply conjugacy to solve problems in the general theory of groups and in specific examples.

■ Produce examples of fields and determine if a given set is a vector space; compute the matrix form of a linear transformation with respect to a fixed bases and relate two bases via a change of basis matrix.

■ Produce examples of groups acting on vector spaces; construct the group of all symmetries of the n-gon.

■ State all definitions and results (lemmata, theorems, corollaries, propositions) covered in lectures, and apply these to solve suitable problems.

■ Prove the results covered in lectures, apply and adapt these proofs in novel situations.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.