### Lecturer

### Timing and Structure

Weeks 4 & 8 Lent Term 1 lecture/week; weeks 5-7 Lent Term 2 lectures/week. 8 lectures

## Aims

The aims of the course are to:

- Introduce the ideas and techniques of Linear Algebra, and illustrate some of their applications in engineering.

## Objectives

As specific objectives, by the end of the course students should be able to:

- For all objectives, complete calculations by hand for small problems, or by using Matlab for larger problems (the IB Computing Course deals with this in detail).
- Solve a set of linear equations using Gaussian elimination, and complete the LU factorisation of a matrix.
- Understand, and be able to calculate bases for the four fundamental subspaces of a matrix.
- Calculate the least squares solution of a set of linear equations.
- Orthogonalize a set of vectors, complete the QR factorisation of a matrix, and be able to use this to find the least squares solution of a set of linear equations.
- Find the eigenvalues and eigenvectors of a matrix, and complete the A = SL S-1 or A = QL QT factorisations as appropriate.
- Find the SVD of a matrix, and to understand how this can be used to calculate the rank of the matrix, and to provide a basis for the each of its fundamental subspaces.

## Content

- Solution of the matrix equation
*Ax = b*: Gaussian elimination,*LU*factorization, the four fundamental subspaces of a matrix. - Least squares solution of
*Ax = b*for an*m x n*matrix with*n*independent columns: Gram-Schmidt orthogonalization,*QR*decomposition. - Solution of
*Ax = l x*, eigenvectors and eigenvalues. - Singular Value Decomposition.

### Booklists

Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.

### Examination Guidelines

Please refer to Form & conduct of the examinations.

### UK-SPEC

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

#### IA1

Apply appropriate quantitative science and engineering tools to the analysis of problems.

#### KU1

Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.

#### KU2

Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.

#### E2

Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.

#### E3

Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.

#### P8

Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.

#### US1

A comprehensive understanding of the scientific principles of own specialisation and related disciplines.

#### US2

A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.

#### US3

An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.

Last modified: 30/05/2023 15:14