University of Wales, Bangor - School of Informatics Mathematics H1 Modules

Lecturer Prof Tim Porter

Recommended book:

Elementary Linear Algebra, H Anton, John Wiley, 1994, 7th edition, ISBN 0471509000, (QA184.A57).

Aims

The main aim of this course is to introduce the student to the importance of linear methods in algebra and to develop the skills and knowledge to apply them to a range of problems.
The secondary aim is to develop basic concepts of numerical calculation and provide a basic understanding of algorithmic ideas for working with two-dimensional arrays, and to introduce simple matrix algorithms for solving systems of simultaneous equations and eigenvalue problems.

Objectives
At the end of the course students should:

• Be able to convert square matrices with distinct eigenvalues to diagonal form.
• Be able to apply algorithms involving row and column operations on a matrix:
  LU-factorisation with partial pivoting; Gauss-Jordan inversion; reduction to Hessenberg form.
• Understand the power methods and the Sturm sequence algorithm for obtaining eigenvalues and eigenvectors of a matrix.
• Be able to modify Maple worksheets which implement these algorithms, and to write Maple procedures to implement related methods.

The course covers the following topics:

• Subspaces of real and complex vector spaces, including the row space, column space and nullspace associated to a matrix.
• Factorisation of matrices using elementary operations, and applications to solving systems of linear differential equations.
• Eigenvalues, eigenvectors and the transformation of a matrix to a similar, diagonal form.
• Inner product spaces. Projections onto subspaces and the method of least squares.
• Algorithms of computational linear algebra, for calculating solutions of simultaneous equations and eigenvalue problems.

Pre-requisites: IAL1032, ICP1024

Syllabus

Matrix factorisation by Elementary Operations
Revision of Gaussian elimination and the solution of systems of homogeneous and non-homogeneous linear equations.
LU-factorisation of a matrix using row operations and partial pivoting. Revision of real vector spaces.
The row, column, and null subspaces associated to a real matrix.
Gauss-Jordan inversion; Hessenberg form; the Smith normal form of an integer matrix.

Diagonalisation, with applications to differential equations
Eigenvectors and eigenvalues: eigenvalues; the characteristic polynomial det(A-xI); Gerschgorin's theorem. The eigenspace of an eigenvalue; diagonalisation criteria.
Iteration: the power, inverse power, and shifted inverse power methods.
Solving systems of linear differential equations using diagonalisation.

Real and Complex Inner Product Spaces
The standard inner product in R^n.
The Gram-Schmidt Process; the QR-factorisation of a matrix.
Orthogonal complement of a subspace. The projection of a vector onto a subspace.
Complex vector spaces and inner product spaces.
The least squares solution of an overdetermined system of linear equations.
Householder transformations and their use in obtaining a Hessenberg form.

Symmetric, Orthogonal, Hermitean and Unitary matrices
Eigenvalues of summetric, skew-symmetric, Hermitean and skew-Hermitean matrices.
Diagonalisation of a real symmetric matrix by an orthogonal matrix, and diagonalisation of an Hermitean matrix by a unitary matrix.
Application to quadratic forms: conics and quadrics.
Sturm sequences and the location of eigenvalues of tridiagonal, Hermitean matrices between consecutive integers.

Assessment:

• Continuous assessment: assignments or tests: 40%
• 1½ hour closed book end of semester examination: 60%

School of Informatics home page Mathematics home page U. W. Bangor home page Latest modification to this page: 11/09/03