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: 2019-20
• See module specification for other years: 2018-19

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Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20 to Spring Term 2019-20

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

• Linear systems of equations in n real or complex variables and their solutions: linear combinations, linear span, subspaces, independence and bases, dimension, existence and uniqueness of solutions to linear systems described through the kernel (null space) and image of a matrix, the Rank-Nullity Theorem.
• Linear transformations: linearity of maps between coordinate spaces, composition as matrix multiplication, one-to-one and onto as consequences of nullity and rank, eigenvectors and eigenvalues, diagonalisability as a  change of coordinates, diagonalisability criteria. Properties of trace and determinant.
• Real and Hermitian inner products: real inner products and symmetric, positive definite matrices; Hermitian inner products and Hermitian symmetric positive definite matrices. Cauchy-Schwarz and triangle inequality. Orthonormal and unitary bases, orthogonal projection onto a subspace, Gram-Schmidt procedure. Diagonalisability of real and Hermitian symmetric matrices.
• Abstract linear algebra: definition of a vector space. All concepts (linear combination, subspace, independence and bases, linear transformations, kernel and image, eigenvectors and eigenvalues) as above with coordinate space replaced by abstract vector space. Additional structure required for real and Hermitian inner products.

• 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.