# Linear Algebra for the Natural Sciences - MAT00041I

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• Department: Mathematics
• Module co-ordinator: Dr. Ian McIntosh
• Credit value: 10 credits
• Credit level: I
• Academic year of delivery: 2018-19
• See module specification for other years: 2019-20

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## Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19 to Spring Term 2018-19

## Module aims

Linear Algebra underpins a very significant part of mathematical modelling, and to be successfully applied in the sciences it is necessary to understand both the theory and the practice of using linear algebra. This module will cover fundamental material and address applications and techniques which students will subsequently be able to draw on in various contexts.

## Module learning outcomes

Subject content

• Definition of abstract vector spaces over R and C: axioms and simple consequences thereof. Connection with R^n and C^n; examples of vector spaces of functions.

• Subspaces

• Linear combinations: the span, linear dependence and linear independence.

• Bases and dimension.

• Co-ordinates: change of co-ordinate matrix when basis is changed.

• Linear transformations: Kernel, Image, characterisation of 1-1 and onto; rank-nullity theorem; isomorphisms; every finite-dimensional vector space is isomorphic to some F^n.

• Co-ordinates again: matrix representation of linear maps; isomorphisms represented by invertible matrices.

• Eigenvalues and eigenvectors: diagonalisation of a linear map on a finite-dimensional space; criterion for diagonalisability in terms of eigenvectors;

• Revision of determinants; characteristic polynomials and eigenvalues.

• Real inner product spaces: length, Cauchy-Schwarz and triangle inequality; orthogonal vectors, orthogonal bases; isometries correspond to orthogonal matrices.

• Hermitian inner product spaces: unitary matrices and results of previous section adapted to this setting; every Hermitian symmetric matrix is diagonalisable by a unitary transformation; simultaneous diagonalisation of commuting Hermitian symmetric matrices.

• Orthogonal subspaces: projections, orthogonal projections and decompositions.

• Use of numerical algorithms (such Gauss-Jordan elimination, QR decomposition) in Linear Algebra, using MATLAB or equivalent software.

• It is hard to overstate the importance of linear algebra in a mathematician’s toolkit. Techniques and results from linear algebra are used across the full spectrum of mathematics and its applications, both in an academic setting and in the wider world. To take an example, as well as having concrete applications in all three of our second year streams, the theory of eigenvalues and eigenvectors is essential in Google’s PageRank algorithm.

## Module content

The theoretical material is developed in the lectures in Autumn term. In Spring term there are 5 practical classes which focus on understanding numerical algorithms and the practical application of linear algebra to mathematical modelling.

## Assessment

Task Length % of module mark
Essay/coursework
Computing Assignment
N/A 20
University - closed examination
Linear Algebra for the Natural Sciences
1.5 hours 80

None

### Reassessment

Task Length % of module mark
Essay/coursework
Computing Assignment
N/A 20
University - closed examination
Linear Algebra for the Natural Sciences
1.5 hours 80

## Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.