Module MATH2021: LINEAR ALGEBRA II

Department: Mathematical Sciences

MATH2021: LINEAR ALGEBRA II

Prerequisites

• Core A Mathematics (MATH1012)

Corequisites

• None.

Excluded Combination of Modules

• Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571), Foundation Mathematics (MATH1641)

Aims

• To develop Linear Algebra (vector spaces, inner product spaces) providing a foundation for study in a number of other areas.

Content

• Vector spaces: Axioms, subspaces, linear independence, spanning sets, bases, dimension, coordinates.
• Linear Mappings: Definition.
• Image, isomorphism, epimorphism, automorphism, endomorphism.
• Rank + nullity = dimension, matrix representation, change of basis, canonical form, determinant.
• Vector space of linear maps, dual space, dual basis.
• Eigenvalues and Eigenspaces: Characteristic polynomial, similar matrices, diagonalisation, Cayley Hamilton theorem, symmetric, skew-symmetric matrices, sesquilinear forms, Hermitian and unitary matrices.
• Sturm-Liouville theory: Special functions as solutions of S-L equation, eigenvalues and eigenfunctions and applications.
• Normal Forms: Endomorphisms.
• Invariant subspaces, minimum polynomial, rational normal form.
• Jordan normal form.

Learning Outcomes

• By the end of the module students will: be able to solve a range of predictable and unpredictable problems in Linear Algebra.
• have a knowledge and understanding of fundamentals of Linear Algebra.
• have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Vector Spaces.
• Linear Mappings.
• Eigenvalues and Eigenspaces.
• Inner Product Spaces.
• Sturm-Liouville Theory.
• Normal Forms.

• In addition students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Abstract reasoning and spatial awareness

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lecturing demonstrates what is required to be learned and the application of the theory to practical examples.
• Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills.
• Tutorials provide active engagement and feedback to the learning process.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Weekly homework problems.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University