# LINEAR ALGEBRA - 2024/5

Module code: MAT1034

## Module Overview

This module is an introduction to Linear Algebra, a fundamental mathematical discipline that revolves around the exploration of vector spaces as well as linear maps between these spaces. This often necessitates the solution of systems of linear equations, which are elegantly described by matrices and vectors. Techniques from Linear Algebra find application in numerous disciplines ranging from pure and applied mathematics to many fields of science, engineering, computing and economics. This module builds on MAT1031: Algebra and provides a foundation for a variety of subsequent modules, including MAT2009 Operations Research and Optimisation, MAT2047 Curves and Surfaces, MAT2048 Groups & Rings, MAT3004 Introduction to Function Spaces and MAT3039 Quantum Mechanics.

### Module provider

Mathematics & Physics

WOLF Martin (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 86

Seminar Hours: 5

Tutorial Hours: 22

Guided Learning: 15

Captured Content: 22

Semester 2

None.

## Module content

Indicative content includes:

• Systems of linear equations: Rank and nullity of matrices. Elementary row operations and row-echelon form. Gaussian elimination. Solubility of linear equations.

• Vector spaces: Axiomatic development of vector spaces. Linear independence of vectors. Basis representations of vectors. Change of basis. Dimension. Vector subspaces.

• Linear operators: Basic properties of linear maps. Rank-nullity theorem. Matrix representation. Cayley-Hamilton theorem. Eigenvalues, eigenvectors and eigenspaces. Algebraic and geometric multiplicities. Diagonalisation.

• Inner product spaces: Inner products and norms. Cauchy-Schwarz inequality. Orthogonality, orthogonal complements and the Gram-Schmidt process. Orthogonal and unitary changes of basis. Isometries. Self-adjoint operators.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 25
Examination End-of-Semester Examination (2 hours) 75

None.

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions and results in Linear Algebra.

• The ability to identify and use the appropriate methods to solve unseen problems relating to linear maps and systems of linear equations.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 25% of the module mark, corresponding to Learning Outcomes 1 and 2.

• A synoptic examination (2 hours), worth 75% of the module mark, corresponding to all Learning Outcomes 1 to 6.

Formative assessment

There are three formative unassessed courseworks over an 11 week period, designed to consolidate student learning.  Students will also receive formative feedback at biweekly seminars on problem sheets, provided to students in advance and designed to consolidate student learning.

Feedback

Students will receive feedback on both the unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from all three courseworks and the in-semester test will assist students with preparation for the synoptic examination. This feedback is complemented by verbal feedback at biweekly seminars and at office hours.

## Module aims

• The aim of this module is to extend students¿ knowledge of vectors, matrices and systems of linear equations, and to introduce them to the abstract concepts of vector spaces, linear maps and inner products.

## Learning outcomes

 Attributes Developed 001 Students will be able to decide if a system of linear equations is soluble and construct solutions via Gaussian elimination. KCT 002 Students will know and be able to apply the definitions of vector spaces and vector subspaces. KC 003 Students will be able to analyse properties of vectors such as linear independence, construct bases of vector spaces, and carry out changes of basis. KC 004 Students will understand the concepts of linear maps and their matrix representations. They will be able to construct eigenvalues and eigenvectors, and diagonalise matrices. KC 005 Students will understand the concepts of inner products, norms and orthogonality. They will be able to carry out orthogonalisation of vectors via the Gram-Schmidt process. KC 006 Students will be able to understand and construct simple proofs similar to those encountered in the module. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Provide students with an introduction to Linear Algebra and its applications.

• Provide students with experience of methods used to understand and solve problems relating to linear maps and systems of linear equations.

The learning and teaching methods include:

• Module notes and videos for students to read through and watch in advance of the weekly tutorials.

• Tutorials. These tutorials provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Seminars for guided discussion of solutions to problem sheets (provided to students in advance) to reinforce their understanding and guide their learning.

• Three unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT1034

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MAT1034 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MAT1034 equips students with skills which significantly enhance their employability. The mathematical proficiency gained hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

Global and Cultural Capabilities: Student enrolled in MAT1034 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in seminars and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT1034 is a module which demands a rigorous approach to linear algebra as well as the ability to perform matrix calculations accurately. Students will gain skills in analysing problems and lateral thinking, and will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.