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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
Lecturer
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.
Textbook

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).
Assessment
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.