General description
In this module the concept of a vector space is introduced. This develops some ideas which have occurred in the first year courses. It will be seen that superficially different problems in mathematics can be unified. For example, the solution of systems of linear equations and linear differential equations are essentially the same process and can be dealt with simultaneously in this context.

Aims
To develop some matrix theory techniques which have occurred in the first year courses in an abstract setting. To introduce the concepts of a vector space and a mapping between vector spaces. To develop further techniques for computation in vector spaces and to study some geometrical applications.

Learning outcomes
On completion of this module, a student should be able to:

• determine whether given algebraic structures are vector spaces;
• apply criteria for subspaces of a vector space;
• determine bases for vector spaces;
• prove propositions in the theory of vector spaces;
• describe the concept of linear transformation;
• calculate matrices representing linear transformations;
• determine the rank and nullity of linear transformations and matrices;
• perform calculations in inner product spaces;
• diagonalise matrices, especially symmetric matrices, and draw inferences concerning the geometrical context.

Syllabus
1. VECTOR SPACES: Definition and examples, subspaces, linear independence, basis and dimensions.
2. LINEAR TRANSFORMATIONS:Definition and exmples, the matrix of a linear transformation, change of basis. The kernel and image of a linear transformation, rank and nullity. The rank of a matrix.
3. INNER PRODUCT SPACES: 3-dimensional geometry, Euclidean and unitary spaces. Orthogonality and Gram-Schmidt orthogonalisation process.
4. DIAGONALISATION OF MATRICES: Eigenvalues and eigenvectors, characteristic equation. Diagonalisation of symmetric matrices, quadratic forms. Applications to geometry, conics and quadrics.