A lecture-based module, open to all students with suitable grounding.  Vectors in geometry are lines with arrows (representing translations in space), added by the parallelogram law.  Vectors in algebra are anything that can be modelled by lines with arrows in geometry, obeying certain rules.  A vector space is all vectors that can be constructed from some given set of vectors using these rules; it has a ‘dimension’.  A low-dimension space can sit inside a high-dimension space as a subspace.  The process of modelling one vector space by another is performed by a ‘linear map’ ; it too obeys certain rules.  Pairs of vectors can be related like forces and distances in Physics, with a dot product representing work done, or a quadratic form representing stored energy.

The aim of linear algebra is to recognise when these models are possible, and to choose the coordinate system to make everything as simple as possible.

• recognise when some system satisfies the rules for a space of vectors, a field of numbers, a linear map between vector spaces, a dot product between vector spaces and so on.
• understand concepts such as dimension, linear independence, linear maps, rank, relations of ‘equivalence’, ‘similarity’, or ‘congruence’ between matrices.
• understand and explain some simple constructions and proofs.
• choose a new coordinate system in a vector space  to simplify the relation between a space and a subspace, or a linear map, or a dot product, or a quadratic form.
• understand how much simplification is or is not possible by choosing a good coordinate system.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

Finding the basis for a vector space or subspace.  Recognising structures by the rules they obey.  Choosing a good coordinate system in various circumstances.

Transferable Skills:

Recognising when a situation can be modelled by vectors with the parallelogram law of addition, so that geometrical ideas can be used.  Ability to change the system of coordinates.

Arguing clearly and rigourously

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Vector spaces
  • Definition and examples of vector space over fields. 
  • Linearly independent sets and spanning sets of vectors. 
  • Subspaces, Bases of coordinates, dimension.

• Linear maps
  • Matrix of a linear map between vector spaces. 
  • Range space (image) and null space (kernel); rank and nullity. 
  • Change of bases.

• Examples of linear maps
  • Vector dot products, dual pairs of spaces, inner product spaces. 
  • Volumes and determinants as multi-linear maps. 
  • Bilinear and quadratic forms. 

• Eigenspaces of linear maps
  • Eigenvalues and eigenvectors of a linear map. 
  • The eigen-polynomial (characteristic polynomial). 
  • Simplifying by changing the basis of coordinates. 
  • Jordan’s normal form.

• Functions of matrices
  • Definition and examples. 
  • Applications to simultaneous ordinary linear differential equations.

• More examples of fields
  • Finite fields of prime order. 

Please see Background Reading List for an indicative list.

Elementary Linear Algebra, Anton, H., Wiley

Linear Algebra, Lang, S., Springer Undergraduate Texts in Mathematics