Linear algebra is one of the foundation stones of mathematics.  The module introduces students to its most fundamental abstract notions: vector spaces, linear subspaces, linear maps, linear independence, and linear spans.  These are accompanied by a plethora of numerical and geometrical examples, firmly establishing the link between the abstract and the concrete.  In particular, the module culminates in the notion of a basis of a vector space, and the demonstration that it identifies every abstract (finite-dimensional) vector space with that of n-tuples of numbers.

• Understand the basic abstract notions of linear algebra: vector space, linear map, basis, etc.
• Understand and work with concrete examples of these abstract notions for the vector spaces of n-tuples of scalars, matrices, polynomials, etc.
• Check a set of vectors for linear independence and/or spanning properties

27 fifty-minute lectures

5 fifty-minute tutorial classes

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 worked solutions for tutorial classes.

Ability to work with vectors and linear maps, and gain an appreciation of their importance within mathematics.

Transferable Skills:
Ability to understand, manipulate and formulate abstract mathematical concepts.  Critical thinking and problem solving skills.  Ability to present work in a scholarly manner.

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

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.

• Introduction and motivation

• Vector spaces over R (and also over Q or C).

• Examples of vector spaces: geometrical vectors, n-tuples or numbers, matrices, polynomials, continuous functions.

• Linear subspaces. Examples of linear subspaces

• Linear maps. Examples of linear maps. Composition.

• The kernel and the image of a linear map.

• Injective and surjective linear maps. Isomorphisms.

• Linear combinations. The span of a set of vectors.

• Linearly independent (LI) sets.

• A basis of vector space. Roll-over Theorem.

• The dimension of a vector space.

• Finite-dimensional vector spaces.Every finite-dimensional vector space over R is isomorphic to Rⁿ

Each point will be illustrated by numerous examples.

Lang, S. 2010. Linear algebra. 3rd ed. Springer 

Anton, H. 2014. Elementary linear algebra. 11th ed. Wiley.