ENGINEERING TRIPOS PART IB - 2011/2012

PAPER 7 - MATHEMATICAL METHODS (2)

Linear Algebra

Leader: Dr J. P. Jarrett

Timing: Weeks 6 Lent term, 2 lectures/week; weeks 7-8 Lent term, 3 lectures/week

Structure: 8 lectures

AIMS

Linear Algebra provides important mathematical tools which are not only essential to solve many technical and computational problems, but also help in obtaining a deeper understanding of many areas of engineering.

The aims of this course are to:

• Introduce the ideas and techniques of Linear Algebra, and illustrate some of their applications in engineering.
• Introduce the ideas and techniques of Contour Integration and illustrate some of their applications in engineering

OBJECTIVES

For all objectives, the student should be able to complete calculations by hand for small problems, or by using Matlab for larger problems (the IB Computing Course deals with this in detail). By the end of the course students should:

• Be able to solve a set of linear equations using Gaussian elimination, and complete the LU factorisation of a matrix;
• Understand, and be able to calculate bases for the four fundamental subspaces of a matrix;
• Be able to calculate the least squares solution of a set of linear equations;
• Be able to orthogonalize a set of vectors, complete the QR factorisation of a matrix, and be able to use this to find the least squares solution of a set of linear equations;
• Be able to find the eigenvalues and eigenvectors of a matrix, and complete the A = SL S^-1 or A = QL Q^T factorisations as appropriate;
• Be able to find the SVD of a matrix, and to understand how this can be used to calculate the rank of the matrix, and to provide a basis for the each of its fundamental subspaces

LECTURE SYLLABUS

• Solution of the matrix equation Ax = b: Gaussian elimination, LU factorization, the four fundamental subspaces of a matrix.
• Least squares solution of Ax = b for an m x n matrix with n independent columns: Gram-Schmidt orthogonalization, QR decomposition.
• Solution of Ax = l x, eigenvectors and eigenvalues.
• Singular Value Decomposition (if time)
• Complet integration, Cauchy's Theorem and Conformal Mapping (if time)

REFERENCES

Please see the Booklist for Part IB Courses for references for this module.

Last updated: August 2011