# Module MA1212: Linear Algebra II

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Hilary term 2012-13
**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:
- Compute the rank of a given linear operator, and use proofs of theoretical results on ranks explained in the course to derive similar theoretical results;
- Compute the dimension and determine a basis for the intersection and the sum of two subspaces of a given space, determine a basis of a given vector space relative to a given subspace;
- Calculate the basis consisting of eigenvectors for a given matrix with different eigenvalues and, more generally, calculate the Jordan normal form and a Jordan basis for a given matrix;
- Apply Gram-Schmidt orthogonalisation to obtain an orthonormal basis of a given Euclidean space;
- Apply various methods (completing the squares, Sylvester's criterion, eigenvalues) to determine the signature of a given symmetric bilinear form;
- Identify the above linear algebra problems in various settings (e.g. in the case of the vector space of polynominals, or the vector space of matices of given size), and apply methods of the course to solve those problems

**Module Content**-
We will cover the following topics, yet not necessarily in the order listed.

- Kernel and image, rank and nullity, dimension formula.
- Characteristic polynomial, eigenvalues and eigenvectors, Jordan form.
- Cayley-Hamilton theorem, minimal polynomial of a linear operator.
- Invariant subspaces, orthogonal complements, direct sums.
- Inner product spaces, orthonormal bases, Gram-Schmidt, Bessel's inequality.
- Bilinear and quadratic forms, Sylvester's criterion, spectral theorem.
- Applications: recurrence relations, least squares approximation.

**Textbook**We will not follow any particular textbook. Two typical references are-
*Algebra*by Michael Artin, -
*Basic linear algebra*by Blyth and Robertson.

Notes, homework assignments and solutions will be posted on the web page

`http://www.maths.tcd.ie/~pete/algebra2` **Module Prerequisite**- MA1111: Linear algebra I
**Assessment Detail**-
This module will be examined in a 2 hour
**examination**in Trinity term. Homework assignments will be due every Thursday. 20% homework, 80% final exam (based on homework and tutorials). The annual**continuous assessment**mark will be carried over and contribute to the final grade for the module at the supplemental examination session.