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# Linear Algebra - MAT00026I

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• Department: Mathematics
• Module co-ordinator: Information currently unavailable
• Credit value: 10 credits
• Credit level: I
• Academic year of delivery: 2018-19

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## Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19

## Module aims

The Core modules in the second year of mathematics programmes cover material which is essential for accessing a wide range of topics later on. Such material underpins the development of mathematics across all three of the major streams we offer through our single-subject degrees, and it is also very important for many combined programmes. As well as covering fundamental material, these modules address applications and techniques which all students will subsequently be able to draw on in various contexts.

As part of the broad aims outlined above, this module develops the theory of linear algebra, starting with the abstract notion of a vector space and developing the tools and techniques needed for a wide range of applications (especially in contexts where matrices are used). As well as applications, this module gives students an opportunity to see the development of a pure mathematical subject from an axiomatic starting point.

## 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.

• 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.

• 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.

## Assessment

Task Length % of module mark
University - closed examination
Linear Algebra
1.5 hours 100

None

### Reassessment

Task Length % of module mark
University - closed examination
Linear Algebra
1.5 hours 100

## 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.