## Mathematics 1

### Module aims

In this module you will have the opportunity to:

• explore the convergence of sequences, infinite sums, and power series
• use convergence to reason about continuity of differentiable functions
• exercise mathematical proofs of lemmas and theorems in order to gain confidence in proof techniques
• use linear algebra in order to formalise and reason about linear systems
• explore the applications of linear algebra in various Computing applications

### Learning outcomes

Upon successful completion of this module you will be able to:

• Prove convergence of sequences and series and determine limit values
• Derive Maclaurin and Taylor series and determine radius of convergence
• Solve systems of linear equations using Gaussian elimination
• Determine matrices representing linear mappings
• Compute Eigenvalues and Eigenvectors for simple matrices and explain their application
• Define projections and rotations in matrix form

### Module syllabus

Part 1: Sequences and series

• Convergence and divergence of sequences and series (arithmetic, geometric, harmonic)
• Comparison test/ absolute convergence
• Cauchy convergence and uniqueness of limits
• Least upper bounds as limits of monotonically increasing sequences
• Taylor's theorem and its applications to solving ordinary differential equations
• Convergence and continuity
• Finite-precision arithmetic and its effect on computation accuracy

Part 2: Linear algebra

• Linear equation systems
• Gaussian elimination
• Linear independence
• Linear mappings
• Vector spaces
• Determinants
• Eigenvalues and diagonalisation
• Intersections of subspaces
• Scalar products and orthogonality
• Projections and rotations

### Teaching methods

The material will be taught through lectures that mix exercises and taught content freely, with an emphasis on interaction, mediated through the use of tools such as Mentimeter. Use of smart pens or whiteboard, in combination with projected material, will be used to explain the key mathematical principles explored. The module is motivated by applications in Computing, such as pagerank, computer graphics and machine learning, with the intention of tying together theory and practice.

There are weekly unassessed, i.e. formative, tutorial exercises that are submitted for marking and feedback as part of separate Maths Methods Tutorials (MMTs), which are weekly one-hour tutorials run by Graduate Teaching Assistants (GTAs) throughout the first term. These tutorials encourage group discussions and group problem solving designed to reinforce your understanding of key topics in continuous mathematics.

The Piazza Q&A web service will be used as an open online discussion forum for the module.

### Assessments

There is one assessed coursework and one assessed test which together contribute 15% of the mark for the module. There is also a written exam which counts for the remaining 85% of the marks for the module.

Detailed feedback, both written and verbal, will be given on the exercises covered in the MMT tutorials. These exercises will be issued every week and written feedback will be returned the week after submission. You will receive written feedback on the assessed coursework exercise, which counts for 15% of the module mark.