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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: 2019-20

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## Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

## 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.

This module develops linear algebra through the study of general properties of linear systems of equations and linear transformations, moving on from the material developed in the first year to make the natural transition from linear properties in coordinate spaces to abstract linear algebra.

## 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.

## Assessment

Task Length % of module mark
Online Exam
Linear Algebra
1.5 hours 100

None

### Reassessment

Task Length % of module mark
Online Exam
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.