Module MA1212: Linear algebra II
- Credit weighting (ECTS)
- 5 credits
- Semester/term taught
- Hilary term 2017-18
- Contact Hours
- 11 weeks, 3 lectures including tutorials per week
- Prof. Paschalis Karageorgis
- Learning Outcomes
- On successful completion of this module, students will be able to:
- find an explicit basis for the null space of a given matrix;
- solve linear recursive relations involving two or more terms;
- apply standard techniques to obtain the Jordan form and a Jordan basis for a given complex square matrix;
- compute the matrix of a bilinear form with respect to a given basis;
- apply various methods (completing the square, Sylvester's criterion, eigenvalues) to find the signature of a symmetric bilinear form;
- combine various results established in the module to either prove or disprove statements involving concepts introduced in the module.
- Module Content
The main concepts to be introduced in this module are the following.
- Diagonalisation: recursive relations, diagonalisable matrix, eigenvalues, eigenvectors, characteristic polynomial, null space, nullity.
- Jordan forms: generalised eigenvectors, column space, rank, direct sum, invariant subspace, Jordan chain, Jordan block, Jordan form, Jordan basis, similar matrices, minimal polynomial.
- Bilinear forms: matrix of a bilinear form, positive definite, symmetric, inner product, orthogonal and orthonormal basis, orthogonal matrix, quadratic form, signature, Sylvester's criterion.
We will not follow any particular textbook. Some typical references are
- Algebra by Michael Artin,
- Matrix theory: a second course by James Ortega,
- Elementary linear algebra with applications by Anton and Rorres.
Notes, homework assignments and solutions will be posted on the web page http://www.maths.tcd.ie/~pete/ma1212.
- Module Prerequisite
- MA1111 (Linear algebra I).
- This module will be examined in a 2-hour examination in the Trinity term. Continuous assessment will count for 20% and the annual exam will count for 80%. Students who are required to take a supplemental exam will be assessed based on that exam alone.