Linear Algebra - 2019 entry

This module focuses on vector spaces and linear systems, giving a rigorous treatment of algebraic techniques. The material of this module underpins several subsequent modules. Building on Vectors and Matrices, this module focuses on further in-depth studies of properties and characterization of vector spaces and manipulation of elements of vector spaces via linear maps, providing you with algebraic techniques, methodologies and some fundamental notions of modern algebra. Linear Algebra provides you with a solid base for your further studies, as it contributes to almost every field and topic within Mathematical Sciences.

This is a necessary foundation for subsequent modules in higher years, including Graphs, Networks and Algorithms, and Dynamical Systems and Control. Topics will include vector spaces, linear transformations, canonical forms and inner product spaces. The theoretical foundations will be complemented by a range of applied examples modelling technological and natural processes, crucially exploiting Linear Algebra to understand and enable these processes.

Pre-requisite modules: “Calculus and Geometry” (ECM1901) and “Vector and Matrices” (ECM1902) or equivalent

This module builds on “Vector and Matrices” (ECM1902). The aim is to advance and extend the concepts building on vectors and matrices, and to introduce theoretical foundations of algebraic theory. Algorithmic aspects and a rigorous theoretical development with proofs of theorems and methodology will be given equal importance.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 develop a basic understanding of general vector spaces and linear maps between vector spaces;

2 develop an understanding of theory and methodologies for matrix transformation and normal forms;

3 demonstrate a basic understanding of applying linear algebra to problems in applied sciences and engineering;

Discipline Specific Skills and Knowledge:

4 demonstrate a clear understanding of fundamental mathematical concepts, manipulations and results of algebraic theory;

5 demonstrate competency in implementation of algebraic methods, and understanding of their relevance within the mathematical sciences and their applications to engineering and science;

Personal and Key Transferable / Employment Skills and Knowledge:

6 reason using abstract ideas;

7 formulate and solve problems and communicate reasoning and solutions effectively in writing and presentation;

8 demonstrate appropriate use of learning resources;

9 demonstrate self- and time-management skills.

- Vector spaces: definitions; subspaces; sums of vector spaces; linear independence; spans; bases and dimension;

- Linear maps and transformations: images and kernels; ranks and nullities; isomorphisms; matrices of linear transformations (composing linear transformations, change of basis matrices); invertibility;

- Eigenvalues and eigenvectors: finding eigenvalues and eigenvectors; the Cayley–Hamilton theorem; diagonalisable linear transformations (direct sums); the minimal polynomial; the Jordan canonical form (finding the Jordan canonical form, finding a Jordan basis, generalised eigenspaces);

- Inner product spaces: inner products; norms; projections; orthonormal bases (Gram-Schmidt); adjoints; Hermitian matrices, real symmetric matrices, unitary matrices, normal matrices and diagonalisability.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic reading:

ELE: http://vle.exeter.ac.uk

Reading list for this module: