# Module MA1112: Linear Algebra II

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2018-19
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
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.

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, 20% midterm exam, 60% final exam (based on homework and tutorials).