Linear algebra is a foundation stone of modern mathematics.  This module will introduce and study many of the fundamental structures of linear algebra, including vector spaces, subspaces, linear transformations, linear combinations, spanning sets, linearly independent sets and dimensionality.  All of these concepts will be illustrated by numerous specific examples, thereby establishing a firm link between the abstract and the concrete.  The module will end with the definition of a basis of a finite-dimensional vector space, and a proof of the result that every finite-dimensional vector space can be identified with a set of finite-tuples of scalars.

• State and apply the vector space axioms.
• Define and understand the basic structures of linear algebra, including vector spaces, subspaces, linear transformations, linear combinations, spanning sets, linearly independent sets, dimensionality and bases.
• Understand and work with specific examples of vector spaces, including geometric vectors, tuples of scalars, matrices and functions.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Appreciation of the general prevalence and significance of linear algebra in mathematics. Ability to understand and apply the fundamental concepts of vector spaces.

Transferable Skills
Recognition of the power of mathematical abstraction and the importance of mathematical rigour. Critical thinking and problem solving skills. Ability to present work in a scholarly manner

• Vectors in three dimensional space.
• Axioms of vector spaces (over the real numbers).
• Examples of vector spaces, including geometric vectors, tuples of real numbers, real matrices and real functions.
• General properties of vector spaces.
• Definition, properties and examples of subspaces.
• Definition, properties and examples of linear transformations.
• Definitions, properties and examples of the kernel and image of a linear transformation, and of isomorphisms.
• Definitions, properties and examples of linear combinations and spanning sets.
• Definitions, properties and examples of linearly dependent and linearly independent sets.
• Definitions, properties and examples of dimensionality and of bases of finite-dimensional vector spaces.​